Analysis III (Lecture)

I followed this course by Prof. Leeb during my3rd_Sem a theLMU. A considerable part of the following notes (and hence of the program of the course) are already present in Analysis II (Lecture), the approach of this lecture was more theoretical (topological).

I. Measure Theory

I.1 Measure Problem

XX

I.2 Measuring Volume with Finite Divisions

I.2.1 Set-theoretic Rings and Algebras

• Def. a set-ring is $\mathcal{R}\subseteq \mathcal{P}(X)$ s.t. it is an (alg.) sub-ring of $(\mathcal{P}(X),\Delta ,\cap )$.
  • for $A\Delta B:=\{x\in X:(x\in A\vee x\in B)\wedge \neg (x\in A\wedge x\in B)\}$, ${\chi }_{\varnothing }\equiv 0$ and ${\chi }_X\equiv 1$.
  • for $\mathcal{R}\subset \mathcal{P}(X)$: $\mathcal{R}$ is a ring $\iff \varnothing \in \mathcal{R}\wedge \mathcal{R}$ is $\setminus$-stable $\wedge \mathcal{R}$ is $\cup$-stable.
    • $A\Delta B=(A\setminus B)\cup (B\setminus A)$, $A\cap B=A\setminus (A\setminus B)$
    • ${\forall }_{A,B\in \mathcal{R}}A\setminus B\in \mathcal{R}\wedge A\cap B\in \mathcal{R}\wedge B\setminus A\in \mathcal{R}$
• Def. a set-alg. $\mathcal{A}$ is a set-ring s.t. $1_{\mathcal{A}}\in \mathcal{A}$.
  • $\mathcal{A}$ is a set-alg. $\iff \varnothing \in \mathcal{A}\wedge \mathcal{A}$ is $\cup$-stable $\wedge \mathcal{A}$ is ${\cdot }^C$-stable
  • ${\forall }_{A,B\in \mathcal{R}}A\setminus B\in \mathcal{R}\wedge A\cap B\in \mathcal{R}\wedge B\setminus A\in \mathcal{R}\wedge (A\cup B{)}^C$
• For $\mathcal{E}\subseteq \mathcal{P}(X)$, def. the set-ring ${\mathcal{R}}_{\mathcal{E}}=\bigcap \{\mathcal{R}:\mathcal{E}\subseteq \mathcal{R}\wedge \mathcal{R}$ is a ring$\}$ on $X$.
• Define ${\mathcal{F}}_0\subseteq ...\subseteq {\mathcal{F}}_n\subseteq ...\subseteq \mathcal{P}(X)$, for ${\mathcal{F}}_0:=\mathcal{E}\cup \varnothing$, ${\mathcal{F}}_n:=\{A\setminus B,A\cup B:A,B\in {\mathcal{F}}_{n-1}\}\supseteq {\mathcal{F}}_{n-1}$; ${\bigcup }_{n\in \mathbb{N}}{\mathcal{F}}_n$.
  • ${\mathcal{R}}_{\mathcal{E}}$ is the smallest ring containing $\mathcal{E}$. Q
• Def. $\mathcal{H}\subseteq \mathcal{P}(X)$ halfr. if (i) $\varnothing \in \mathcal{H}$, (ii) $\mathcal{H}$ is $\mathcal{H}$-st., (iii) ${\forall }_{A,B\in \mathcal{H}}{\exists }_{C_1,...,C_n\in \mathcal{H}}A\setminus B=C_1\bigsqcup ...\bigsqcup C_n$.
  • $\mathcal{R}$ is an hafr.
  • Principal Exampe, (PE) $\mathcal{I}=\{[a,b):a\le b\wedge a,b\in \mathbb{R}\}$ is halfr. on $\mathbb{R}$.
  • (Simultane Zerlegung) ${\forall }_{H_1,...,H_n\in \mathcal{H}}{\exists }_{H_1^{\prime },...,H_n^{\prime }\in \mathcal{H}}{H}_i=H_j^{\prime },...,H_{j^{\prime }}^{\prime }$
  • $\mathcal{R}(\mathcal{H})$ def. Q
• Products
  • ${\mathcal{F}}_1\ast ...\ast {\mathcal{F}}_n:=\{M_1\times ...\times M_n:{\forall }_{i\le n}{M}_i\in {\mathcal{F}}_i\}\subseteq \mathcal{P}(X_1\times ...\times X_n)$, $\cup$-st. cover. for ${\mathcal{F}}_i\subseteq \mathcal{P}(X)$,
    • squares
  • ${\mathcal{F}}_1⊠...⊠{\mathcal{F}}_n:=\{$finite union of subsets of ${\mathcal{F}}_1\ast ...\ast {\mathcal{F}}_n\}$
    • union of squares
  • $\mathcal{Z}({\mathcal{F}}_1,...,{\mathcal{F}}_n):=\{{\pi }_k^{-1}(M_k):M_k\in {\mathcal{F}}_k,1\le k\le n\}$ where ${\pi }_k^{-1}:X_1\times ...\times X_n\to X_k$ is a proj.
    • Cylinder
  • ${\mathcal{H}}_1\ast ...\ast {\mathcal{H}}_n$ is an halfr. on $X_1\times ...\times X_n$, (prod. of halfr.)
  • ${\mathcal{H}}_1⊠...⊠{\mathcal{H}}_n=\mathcal{R}({\mathcal{H}}_1)⊠...⊠\mathcal{R}({\mathcal{H}}_n)$
  • (PE) $[a,b):=[a_1,b_1)\times ...\times [a_d,b_d)$ elem. of $Q^d=\mathcal{I}\ast ....\ast \mathcal{I}\subset \mathcal{P}({\mathbb{R}}^d)$ halfr., $\mathcal{R}(Q^d)={\mathcal{F}}^d:=\mathcal{I}⊠...⊠\mathcal{I}\subset \mathcal{P}({\mathbb{R}}^d)$ (figures).
    • (Zerl. Lemma) every figure is a disj. union of finitely many squares
    • Any finite union of squares is also a figure

I.2.2 Content

• $\mu :\mathcal{H}\to [0,\infty ]$ s.t. (i) $\mu (\varnothing )=0$, (ii) $\mu (A_1\bigsqcup ...\bigsqcup A_n)=\mu (A_1)+...+\mu (A_n)\wedge A_1\bigsqcup ...\bigsqcup A_n\in \mathcal{H}$ (cont.)
• (PE) ${\lambda }_{\mathcal{I}}:Q^1\to [0,\infty ),[a,b)\mapsto b-a$ (euclidian lenght)
• ${\forall }_{A,B\in \mathcal{H}}A\subseteq B\to \mu (A)\le \mu (B)$ (monotony)
• ${\forall }_{A_1,...,A_n\in \mathcal{H}}\mu (A_1\cup ...\cup A_n)\le \mu (A_1)+...+\mu (A_n)$ (subadditivity)
• For $\mu :\mathcal{H}\to [0,\infty ]$ ex. uniq. $\overline{\mu }:\mathcal{R}(\mathcal{H})\to [0,\infty ]$.
  • How to extend the content? Q
• (PE) ${\lambda }_{Q_1}^1:Q^1⟶[0,\infty ],[a,b)\mapsto b-a$ extends to ${\lambda }_{{\mathcal{F}}^1}^1:{\mathcal{F}}^1⟶[0,\infty ]$.
• Products
  • If $({\mu }_1\times ...\times {\mu }_n)({\mathcal{H}}_1\times ...\times {\mathcal{H}}_n)={\mu }_1(H_1)\cdot ...\cdot {\mu }_n(H_n)$, ${\mu }_1\times ...\times {\mu }_n$ cont. ${\mathcal{H}}_1\ast ...\ast {\mathcal{H}}_n$
    • convention: $0\cdot \infty =0$
  • (PE) ${\lambda }_{Q^d}^d:={\lambda }_{Q^1}^1\times ...\times {\lambda }_{Q^1}^1$ ($d$-times) on $Q^d=Q^1\ast ...\ast Q^1\subset \mathcal{P}({\mathbb{R}}^d)$
  • ${\mu }_1⊠...⊠{\mu }_n$ uniq. cont. on ${\mathcal{R}}_1⊠...⊠{\mathcal{R}}_n\subset \mathcal{P}(X_1\times ...\times X_n)$ s.t.:
    • $({\mu }_1⊠...⊠{\mu }_n)(M_1\times ...\times M_n)={\mu }_1(M_1)\cdot ...\cdot {\mu }_n(M_n)$,
      • recall: $(M_1\times ...\times M_n)\in {\mathcal{R}}_1\ast ...\ast {\mathcal{R}}_n$ for $M_i\in {\mathcal{R}}_i$
  • (PE) ${\lambda }_{{\mathcal{F}}^d}^d$ is unique

I.3 Measuring Volume with Countable Divisions

I.3.1 Pre-measures

• def. Q
• $A_n\nearrow A:={\bigcup }_{n=1}^{\infty }{A}_n=A$ for $(A_i{)}_{i\in [1,n]}$ a $\subseteq$-chain (inside approximation), similarly $A_n\swarrow A$.
• let $A_n\nearrow A\wedge A_n^{\prime }\swarrow A$ then ${\lim }_{n\to \infty }\mu (A_n)\le \mu (A)\le {\lim }_{n\to \infty }\mu (A_n^{\prime })$.
• Products
  • ${\mu }_1⊠...⊠{\mu }_n$ pmeas.

I.3.2 $\sigma$-Algebras

• $\mathcal{A}\subset \mathcal{P}(X)$ is $\sigma$-al. if: (i) $\varnothing \in \mathcal{A}$, (ii) $(A_n{)}_{n\in \mathbb{N}}\subseteq \mathcal{A}\Rightarrow {\bigcup }_{n\in \mathbb{N}}{A}_n\in \mathcal{A}$ ($\sigma$-$\cup$-st.), (iii) ${\cdot }^C$-st.
  • eq. to: (i)* $X\in \mathcal{A}$, (ii)* $(A_n{)}_{n\in \mathbb{N}}\subseteq \mathcal{A}\Rightarrow {\bigcap }_{n\in \mathbb{N}}{A}_n\in \mathcal{A}$ ($\sigma$-$\cap$-st.), (iii) ${\cdot }^C$-st.
• By transf. ind. define $\sigma (\mathcal{E}):={\mathcal{E}}_{{\omega }_1}$:
  • for $A\subseteq \mathcal{P}(X)$, $A^{\vartheta }:=$ Q
  • ${\mathcal{E}}_0:=\mathcal{E}\cup \{\varnothing \}$
  • ${\mathcal{E}}_n:={\mathcal{E}}_{n-1}^{\vartheta }$
  • ${\mathcal{E}}_{\lambda }:=({\bigcup }_{n<\lambda }{\mathcal{E}}_n{)}^{\vartheta }$ for $\lambda$ limit ordinal
• Topological Origin
  • $\mathcal{B}(X):=\sigma (\mathcal{T})$ for $(X,\mathcal{T})$ top. sp.
    • (PE) ${\mathcal{B}}^d:=\mathcal{B}({\mathbb{R}}^d)$
      • ${\mathcal{B}}^d=\sigma (A)$ for any of the following options
        • $A$ are open subsets of $X$, i.e. the topology.
        • $A$ are the closed subsets of $X$.
        • $A$ are the compact subsets of $X$.
        • $A$ are the open metrical balls
        • $A$ are the closed metrical balls
        • $A$ are the open rectangles
        • $A$ are the half-open rectangles, i.e. $Q^d$
      • ${\mathcal{B}}^d=\sigma (Q^d)$
• Dynkin Systems
  • $\mathcal{D}\subseteq \mathcal{P}(X)$ is dyn. if (i) $\varnothing \in \mathcal{D}$, (ii)$(A_n{)}_{n\in \mathbb{N}}\subseteq \mathcal{D}\Rightarrow {⨆}_{n\in \mathbb{N}}{A}_n\in \mathcal{D}$, (iii) is ${\cdot }^C$-st.
  • $\mathcal{A}$ is dyn. and if $\mathcal{D}$ is $\cap$-st. it is $\sigma$-al.
  • ${\forall }_{D_1,D_2\in \mathcal{D}}{D}_1\subseteq D_2\to D_2\setminus D_1\in \mathcal{D}$
  • $\mathcal{D}{|}_D:=\{M\subset D|M\in \mathcal{D}\}\subset \mathcal{P}(D)$ (Spur) is dyn on $D$.
  • ${\mathcal{D}}_D:=\{M\subset X|M\cap D\in \mathcal{D}\}\subset \mathcal{P}(X)$ is dyn on $X$.
  • $\mathcal{E}$ $\cap$-st., $\mathcal{D}(\mathcal{E})$ is $\sigma$-al.
• The Measurable Category
  • XX
• Products
  • ${\mathcal{A}}_1\otimes ...\otimes {\mathcal{A}}_n=\sigma (Z({\mathcal{A}}_1,...,{\mathcal{A}}_n))$ Q
    • $Z({\mathcal{A}}_1,...,{\mathcal{A}}_n)=\{{\pi }_k^{-1}(M_k)|M_k\in {\mathcal{A}}_k\wedge k\in [1,n]\}$

I.3.3 Measures

• $\mu :\mathcal{A}\to [0,\infty ]$ is a meas.
  • eq. $\mu :\mathcal{A}\to [0,\infty ]$ a meas. on $\mathcal{A}$ if (i) $\mu (\varnothing )=0$, (ii) $\mu ({⨆}_{n\in \mathbb{N}}{A}_n)={\sum }_{n\in \mathbb{N}}\mu (A_n)$
• External Measures
  • ${\mu }^{\ast }(M):=inf\{{\sum }_{n=1}^{\infty }\mu (A_n):(A_n{)}_{n\in \mathbb{N}}\subseteq \mathcal{H}\wedge {\bigcup }_{n=1}^{\infty }{A}_n\supseteq M\}$ for $\mu$ cont.
    • if empty ${\mu }^{\ast }(M)=inf(\varnothing )=\infty$
    • it is enough to look at the p.disj. $(A_n{)}_{n\in \mathbb{N}}$.
    • $\overline{{\mu }^{\ast }}={\mu }^{\ast }$
  • ${\mu }^{\ast }:\mathcal{P}(X)\to [0,\infty ]$ ext. meas iff
    • ${\mu }^{\ast }(\varnothing )=0$
    • ${\forall }_{M,M^{\prime }\subseteq X}M\subseteq M^{\prime }\to {\mu }^{\ast }(M)\le {\mu }^{\ast }(M^{\prime })$ (monotony)
    • $(M_n{)}_{n\in \mathbb{N}}\subseteq \mathcal{P}(X)$, ${\mu }^{\ast }({\bigcup }_{n=1}^{\infty }{M}_n)={\sum }_{n=1}^{\infty }{\mu }^{\ast }(M_n)$ ($\sigma$-subadditivity)
  • Similarly define the internal measure, ${\mu }_{\ast }$
• Motivation
  • ${\mu }^{\ast }(M)-{\mu }_{\ast }(M)={\mu }^{\ast }(M)+{\mu }^{\ast }(R-M)-{\mu }^{\ast }(R)\ge 0$, bound. $R$ of a fuzzy shape $M$ (discr.)
  • call $M$ meas. if ${\forall }_{S\subset X}{\mu }^{\ast }(S)={\mu }^{\ast }(S\cap M)+{\mu }^{\ast }(S\cap M^C)$.
  • ${\mu }^{\ast }(N)=0$ (nullset)
• Extensions of Pre-measures to Measures
  • (Carathéodory, 1914) ${\mathcal{A}}^{\ast },{\mu }^{\ast }{|}_{{\mathcal{A}}^{\ast }}$ are resp. a $\sigma$-al. and a meas. Q
    • ${\mathcal{A}}^{\ast }:=\{M\subseteq X:M$ is ${\mu }^{\ast }$-meas$\}$.
    • (i) $\sigma (\mathcal{R})\subseteq {\mathcal{A}}^{\ast }$
    • (ii) ${\mu }^{\ast }{|}_{\sigma (\mathcal{R})}$ is a meas.
      • (i), (ii) $\Rightarrow {\mu }^{\ast }{|}_{\mathcal{R}}$ prem., also ${\mu }^{\ast }{|}_{\mathcal{R}}\le \mu$ (recall $dom(\mu )=\mathcal{R}$)
    • $\mathcal{H}\subseteq \mathcal{P}(X)$, $\mu :\mathcal{H}\to [0,\infty ]$ meas., then $\mu$ is finite if $\mu (X)<\infty$ (finite)
  • $\mathcal{H}\subseteq \mathcal{P}(X)$, $\mu :\mathcal{H}\to [0,\infty ]$ meas., it is $\sigma$-fin. if ex $(H_n{)}_{n\in \mathbb{N}}$ s.t. ${\forall }_{x\in X}{\exists }_{n\in \mathbb{N}}x\in H_n\wedge \mu (H_n)<\infty$
  • (Extension Theorem):
    • (Existence) $\mu :\mathcal{R}\to [0,\infty ]$ a prem., it def. a meas. ${\mu }^{\ast }$ on $\sigma (\mathcal{R})$.
    • (Maximality of ${\mu }^{\ast }{|}_{\mathcal{R}}$) for any meas. $\nu$ s.t. $\nu {|}_{\mathcal{R}}=\mu$, $\nu \le {\mu }^{\ast }{|}_{\mathcal{R}}$
    • (Weak Uniq.) $\mu :\mathcal{R}\to [0,\infty ]$ $\sigma$-fin. prem., then there is a uniq. ext to a meas. $\sigma (\mathcal{R})$
      • (PE) ${\lambda }^d:=({\lambda }_{{\mathcal{F}}^d}^d{)}^{\ast }$
• Approximation of Subsets
  • $M$ is ${\mu }^{\ast }$-meas. $\iff {\mu }^{\ast }(\partial M)=0\iff {\mu }^{\ast }(I)={\mu }^{\ast }(A)$
    • $I:=M-\partial M$
    • $A:={\bigcap }_{n=1}^{\infty }{A}_{\frac{1}{n}}\in \sigma (\mathcal{R})$, for ${\forall }_{ϵ>0}{\exists }_{A_{ϵ}\in \sigma (\mathcal{R})}(M\subset A_{ϵ}\wedge {\mu }^{\ast }(A_{ϵ})<{\mu }^{\ast }(M)+ϵ$
  • $\mu$ is $\sigma$-fin. $\Rightarrow {\mathcal{A}}^{\ast }\subseteq \{\alpha :\alpha ={\beta }_1\cup ...\cup {\beta }_n\cup {\gamma }_1\cup ...\cup {\gamma }_m\wedge {\forall }_{i\in [1,max(n,m)]}{\beta }_i\in \sigma (\mathcal{R})\wedge {\mu }^{\ast }({\gamma }_i)=0\}$.
  • $\mu$ is compl. if all $\mu$-nullsets are in $\mathcal{A}$.
    • $\mathcal{N}:=\{N\subseteq X:{\exists }_{N_{\nu }\in \mathcal{A}}N\subseteq N_{\nu }\wedge \nu (N_{\nu })=0\}$
  • $\tilde{\mathcal{A}}:=\{A\cup N:A\in \mathcal{A}\wedge N\in \mathcal{N}\}=\{\tilde{A}\subset X:{\exists }_{A^{\prime }\in \mathcal{A}}\tilde{A}\Delta A^{\prime }\in \mathcal{N}\}\subseteq \mathcal{A}$
    • $\tilde{\mathcal{A}}$ is a $\sigma$-al. and ex a uniq. ext $\tilde{\mu }$.
• More on the Measurable Category
  • Pushforward (Wiki): for $f:X_1\to X_2$ meas., $(X_1,{\mathcal{A}}_1)$, $(X_2,{\mathcal{A}}_2)$ meas. sp. and $\mu :{\mathcal{A}}_1\to [0,\infty ]$ then the pushf. is the meas $f_{\ast }(\mu ):{\mathcal{A}}_2\to [0,\infty ],A_2\mapsto \mu (f^{-1}(A_2))$.
• Product Measure
  • for $(X_i,{\mathcal{A}}_i,{\mu }_i{)}_{i\in [1,n]}$, recall ${\mathcal{A}}_1\otimes ...\otimes {\mathcal{A}}_n\subseteq {\mathcal{A}}_1⊠...⊠{\mathcal{A}}_1\subseteq {\mathcal{A}}_1\ast ...\ast {\mathcal{A}}_n\subseteq Z({\mathcal{A}}_1,...,{\mathcal{A}}_n)$.
    • ${\mathcal{A}}_1\otimes ...\otimes {\mathcal{A}}_n$ is the $\sigma$-al., the other are equivalent generators.
  • def. for $(X_i,{\mathcal{A}}_i,{\mu }_i{)}_{i\in [1,n]}$ $\sigma$-fin. then ${\mu }_1\otimes ...\otimes {\mu }_n$ is the product measure and $(X\times ...\times X,{\mathcal{A}}_1\otimes ...\otimes {\mathcal{A}}_n,{\mu }_1\otimes ...\otimes {\mu }_n)$

I.3.4 Lebesgue Measure

• Topological Approximation of Measurable Sets
  • XX
• Invariances and Behave under Affine Functions
  • for $\varphi :{\mathbb{R}}^d\to {\mathbb{R}}^d$ aff ${\varphi }^{\ast }({\lambda }^d)=|\frac{1}{det(A)}|\cdot {\lambda }^d$

II. Integration Theory

II.1 Integration & Numeral Functions

• a num.f, is $f:X\to \overline{\mathbb{R}}$ for $X$ a set s.t. $(X,\mathcal{A})$ meas.sp. and $\overline{\mathbb{R}}:=\{-\infty ,\infty \}\cup \mathbb{R}$.
  • $\overline{\mathcal{B}}:=\mathcal{B}(\overline{\mathbb{R}})$ is a $\sigma$-al. generated by:
    • $\{[-\infty ,a):a\in \mathbb{R}\}$, also for $(-\infty ,a],[-\infty ,a]$
  • $f:X\to \overline{\mathbb{R}}$ num is meas. $\iff {\forall }_{a\in \mathbb{R}}\{f\le a\}$
    • $\{f\le a\}:=f^{-1}([-\infty ,a])\in \mathcal{A}$
    • \mathcal{M}_\overline{\mathbb{R}}(X, \mathcal{A}) are the meas.num.f. on $(X,\mathcal{A})$
      • closed under $sup$, $limsup$.
    • ${\mathcal{M}}_{\mathbb{R}}(X,\mathcal{A})$ are the meas.num.f. on $(X,\mathcal{A})$ s.t. $im(f)=\mathbb{R}$
    • ${\mathcal{M}}_{[0,\infty ]}(X,\mathcal{A})$ are the meas.num.f. on $(X,\mathcal{A})$ s.t. $im(f)=[0,\infty ]$
• Approximation trough Step Funcitons
  • for $f$ step.f. $f$ meas. $\iff {\forall }_{i\in I}{c}_i$ meas. i.e. each level is meas.
    • $\Rightarrow {\mathcal{T}}_{\mathbb{R}}={\mathcal{T}}_{\mathbb{R}}(X,\mathcal{A}):=$ meas.step.f. s.t. $im(f)=\mathbb{R}$
    • $\Rightarrow {\mathcal{T}}_{[0,\infty ]}={\mathcal{T}}_{[0,\infty ]}(X,\mathcal{A}):=$ meas.step.f. s.t. $im(f)=[0,\infty ]$
  • any $f\in {\mathcal{M}}_{[0,\infty ]}$ is a monotone limit of $t_n\subseteq {\mathcal{T}}_{[0,\infty ]}$: $t_n\nearrow f$
• Regions below Graphs
  • for $f:X\to \mathbb{R}$, f \in \mathcal{M}_\overline{\mathbb{R}} \Leftrightarrow \{(x, y) \in X \times \mathbb{R}: y \le f(x)\} \in \mathcal{A} \otimes \mathcal{B}^1 (prod.$\sigma$-al. of $X\times \mathbb{R}$)
  • $f_1,f_2\in {\mathcal{M}}_{\overline{\mathbb{R}}}\Rightarrow (f_1,f_2{)}^{-1}(A_1\times A_2):=f_1^{-1}(A_1)\cap f_2^{-1}(A_2)\in \mathcal{A}$
    • $f_1+f_2$ (pointwise) is uniq. and meas. (addition)
    • Q (moltiplication)

II.2 Lebesgue Integral

• $f:X\to \overline{\mathbb{R}}$, prod.meas. is $\mu \otimes {\beta }^1=(\mu ⊠{\mathcal{B}}^1{)}^{\ast }{|}_{\mathcal{A}\otimes {\mathcal{B}}^1}$

II.2.1 Integral for Non-negative Functions

• the Integral $I:{\mathcal{M}}_{[0,\infty ]}(X,\mathcal{A})\to [0,\infty ],f\mapsto {\int }_{X}fd\mu :=(\mu \otimes {\beta }^1)(\{(x,y)\in X\times \mathbb{R}:0<y<f(x)\})$
  • $f\le g\Rightarrow {\int }_{X}fd\mu \le {\int }_{X}gd\mu$ (monotony of $\mu \otimes {\beta }^1$)
  • $f_n\nearrow f\in {\mathcal{M}}_{[0,\infty ]}(X,\mathcal{A})\Rightarrow {\int }_X{f}_{n}d\mu \nearrow {\int }_{X}fd\mu$ (monotone convergence, B. Levi)
• for $(f_n{)}_{n\in \mathbb{N}}\subseteq M_{[0,\infty ]}(X,\mathcal{A})$, then $\int {\lim }_{n\to \infty }\inf f_{n}d\mu \le {\lim }_{n\to \infty }\inf {\int }_X{f}_{n}d\mu$ (Fatou’s lemma)
• Linearity & Homogenity
  • ${\int }_X{\chi }_{A}d\mu =\mu (A)$
  • ${\int }_{X}td\mu ={\sum }_{i=1}^n\mu (A_i)a_i$ for $t={\sum }_{i=1}^n{a}_i{\chi }_{A_i}$, ${\forall }_{i\in [1,n]}{A}_i\in \mathcal{A}$, also if $A_i$ are not disj.
  • any $f_{+},f_{-}\in {\mathcal{M}}_{[0,\infty ]}(X,\mathcal{A})$ can be approximated by some $t_{+,n},t_{-,n}$.
  • $I$ has monotony, is additive and pos. homogene.

II.2.2 Integral on Measurable Functions

• for f \in \mathcal{M}_\overline{\mathbb{R}}(X, \mathcal{A}), is $\mu$-int. $\iff {\int }_X|f|d\mu <\infty \iff {\int }_x{f}_{\pm }d\mu <\infty$
  • $|{\int }_{X}fd\mu |\le {\int }_X|f|d\mu$
• $f=g$ alm.ever. if $\mu (\{x:f(x)\ne g(x)\})=0$
  • $f=g$ alm. ever. $\Rightarrow {\int }_{X}fd\mu ={\int }_{X}gd\mu$
  • f \in \mathcal{M}_\overline{\mathbb{R}}(X, \mathcal{A}) is $\mu$-int. in $A$ ($\in \mathcal{A}$) if ${\chi }_A\cdot f$ is $\mu$-int.
• ${\mathcal{L}}_{\mathbb{R}}^1⊊{\mathcal{L}}_{\overline{\mathbb{R}}}^1$ is te VS of meas. func.
  • $f_1,f_2\in {\mathcal{L}}^1\Rightarrow f_1+f_2\in {\mathcal{L}}^1$ (additivity)
  • $f_1,f_2\in {\mathcal{L}}^1\wedge f_1\le f_2\Rightarrow {\int }_X{f}_{1}d\mu \le {\int }_X{f}_{2}d\mu$ (monotony)
• Gemetrical Structure of ${\mathcal{L}}^1$
  • the functional $||\cdot |{|}_1:{\mathcal{L}}^1(X,\mathcal{A},\mu )\to [0,\infty ),f\mapsto {\int }_X|f|d\mu$ is a halfnorm
  • $|f+g|\le |f|+|g|\Rightarrow ||f+g|{|}_1\le ||f|{|}_1+||g|{|}_1$ (triangle inequality)
  • $|{\int }_{X}fd\mu \le ||f|{|}_1$
  • hence ${\mathcal{L}}^1$ is:
    • $\mathbb{R}$-linear
    • monoton
    • Lipzisch cont. for $||\cdot |{|}_1$
  • $f=g$ alm.ever. $\iff ||f-g|{|}_1=0$
  • $\{f\in {\mathcal{L}}^1:||f|{|}_1=0\}$ is a subVS of ${\mathcal{L}}^1$, because of the triangle-ineq.

II.2.3 Convergence Theorems

• $f_n{\to }^{L^1}f\iff ||f_n-f|{|}_1\to 0$
  • $f_n{\to }^{L^1}f\iff {\int }_x|f_n-f|d\mu \to 0\Rightarrow {\int }_x{f}_{n}d\mu \to {\int }_{X}fd\mu \iff {\int }_X(f_n-f)d\mu \to 0$
• $(f_n{)}_{n\in \mathbb{N}}\subseteq {\mathcal{M}}_{\overline{\mathbb{R}}}$, f \in\mathcal{M}_\overline{\mathbb{R}} s.t. $f_n\to f$ $\mu$-alm.ever., then (i) ex. $g\in {\mathcal{L}}_{[0,\infty ]}^1$ s.t. $|f_n|\le g$ alm.ever., (ii) f_n, f \in \mathcal{L}^1_\overline{\mathbb{R}}, (iii) $f_n{\to }^{L_1}f$, (iv) ${\int }_X{f}_{n}d\mu \to {\int }_{X}fd\mu$ (majorant covergence, Lebesgue)
• any Causchy-seq. $(f_n{)}_{n\in \mathbb{N}}\subseteq {\mathcal{L}}_{\overline{\mathbb{R}}}^1$, $f\in {\mathcal{L}}^1$ s.t. $(f_n{)}_{n\in \mathbb{N}}{\to }^{L_1}f$, then ex. a subseq. $(f_{n_k}{)}_{k\in \mathbb{N}}$ s.t. $f_{n_k}\to f$ pointwise $\mu$-alm.ever. (inverse direction)
• $({\mathcal{L}}^1(X,\mathcal{A},\mu ),||\cdot |{|}_1)$ is complete, hence a Banach Space (Riesz-Fischer for $L_1$) Q

II.2.4 Integration over Products

• recall: $({\mu }_1\otimes {\mu }_2)(A_1\times A_2)={\mu }_1(A_1)\cdot {\mu }_2(A_2)$
• If $M\in {\mathcal{A}}_1\otimes {\mathcal{A}}_2$, then ${\forall }_{x_1\in X_1}{M}_{x_1}\in {\mathcal{A}}_2$ and the num.f. $X_1\to [0,\infty ],x_1\mapsto {\mu }_2(M_{x_1})]$ is meas. (i.e. is in ${\mathcal{M}}_{[0,\infty ]}(X,{\mathcal{A}}_1)$ and $({\mu }_1\otimes {\mu }_2)(M)={\int }_{X_1}{\mu }_2(M_{X_1})d{\mu }_1(x_1)$).
  • $\{x_1\}\times M_{x_1}:=M\cup (\{x_1\}\times X_2)$
• Q (Cavalieri Principle)
• $f\in {\mathcal{M}}_{[0,\infty ]}(X_1\times X_2,{\mathcal{A}}_1\otimes {\mathcal{A}}_2)$, then (i) $x_1\mapsto {\int }_{X_2}f(x_1,\cdot )d{\mu }_2\in {\mathcal{M}}_{[0,\infty ]}(X_1,{\mathcal{A}}_1)$, i.e. is meas., (ii) ${\int }_{X_1\times X_2}fd({\mu }_1\otimes {\mu }_2)={\int }_{X_1}({\int }_{X_2}f(x_1,\cdot )d{\mu }_2)d{\mu }_1(x_1)$ (Fubini)
• $f\in {\mathcal{M}}_{[0,\infty ]}(X_1\times X_2,{\mathcal{A}}_1\otimes {\mathcal{A}}_2)\iff x_1\mapsto {\int }_{X_2}f(x_1,\cdot )d{\mu }_2$ is ${\mu }_2$-int. $\Rightarrow f(x_1,\cdot )$ is ${\mu }_2$-int. for $x_1\in X$ ${\mu }_1$-alm.ever.

II.2.5 Transformation of Measures and Integrals

• Transformation of Integrals under meas. Functions
  • for $\phi :(X,\mathcal{A})\to (Y,\mathcal{B})$ meas., define $\phi :(X,\mathcal{A})\to (Y,\mathcal{B})$ (pullback)
  • $(\mathcal{A}{\to }^{\mu }[0,\infty ])\mapsto ({\phi }_{\ast \mu }:\mathcal{B}\to [0,\infty ],M\mapsto \mu ({\phi }^{-1}(M))(={\phi }_{\ast \mu }(M))$ (trans. forward of e meas)
  • (Transformation formulae for integrals under meas. functions)
    • (i) for $f\in {\mathcal{M}}_{[0,\infty ]}(Y,\mathcal{B})$ holds ${\int }_{Y}fd({\phi }_{\ast \mu })={\int }_X(f\circ \phi )d\mu$.
    • (ii) for $f\in {\mathcal{M}}_{\bar{\mathbb{R}}}(Y,\mathcal{B})$, then $f$ is ${\phi }_{\ast }\mu$-int. $\iff f\circ \phi$ $\mu$-int
      • hence ${\int }_{Y}fd({\phi }_{\ast }\mu )={\int }_X(f\circ \phi )d\mu$
• Transformation of the Lebesgue Measure under Diffeomorphims
  • recall: for ${\phi }_{A,r}$ aff. ${\phi }_{A,r}:{\mathbb{R}}^d\to {\mathbb{R}}^d,x\mapsto Ax+r$, then $({\phi }_{A,r}{)}_{\ast }{\beta }^d=|detA{|}^{-1}{\beta }^d$
  • for $C^1$-diffeo. $\phi :U\to V$ op. subset of ${\mathbb{R}}^d$, holds ${\phi }_{\ast }({\beta }^d{|}_U)=|det(d({\phi }^{-1}))|\cdot {\beta }^d{|}_V$.
  • for $\phi$ a $C^1$-diffeo. $\phi :U\to V$ op. subset of ${\mathbb{R}}^d$, contain Borel-Nullsets and is Lebesgue-meas. Also ${\phi }_{\ast }({\lambda }^d{|}_U)=|det(d({\phi }^{-1}))|\cdot {\lambda }^d{|}_V$
• Transformation of Integrals under Diffeomorphisims
  • for $\phi$ a $C^1$-diffeo. $\phi :U\to V$ op. subset of ${\mathbb{R}}^d$: (Transf. for integrals under diffeos)
    • for $f\in {\mathcal{M}}_{[0,\infty ]}(V,\mathcal{B}(V))$:
      • (i) ${\int }_{V}fd{\lambda }^d={\int }_U(f\circ \phi )\cdot |det(d(\phi ))|d{\lambda }^d$
    • for $f\in {\mathcal{M}}_{\bar{\mathbb{R}}}(V,\mathcal{B}(V))$:
      • (i)
      • (ii) $f$ is ${\lambda }^d$-int. $\iff (f\circ \phi )\cdot det(d(\phi ))|$ is ${\lambda }^d$-int.

III. Manifolds, Differential Forms and Stokes’ Theorem

Most of the materialis covered by Youtube: Bright Side, I take notes on these topics here: make the link.

III.1 Differentiable Manifolds

• $M\subseteq {\mathbb{R}}^n$ is $m$-dim (for $m\le n$) ${\mathcal{C}}^k$-submnf. $\iff {\forall }_{p\in M}{\exists }_{\phi :U\to V}\phi (M\cap U)=({\mathbb{R}}^m\times \{0\})\cap V$
  • ${\mathbb{R}}^m\times \{0\}\cong {\mathbb{R}}^m$
  • $\phi$ is the extr.ch. for $M$ in $p$, $k:=\phi {|}_{M\cap U}$ is the intr.ch. in $p$
• Transfer Chart
  • extr.ch. ${\phi }_2\circ {\phi }_1^{-1}:{\phi }_1(U_1\cap U_2)\to {\phi }_2(U_1\cap U_2)$, ${\mathcal{C}}^k$-diffeo.
  • intr.ch. $k_2\circ k_1^{-1}:k_1(M\cap U_1\cap U_2)\to k_2(M\cap U_1\cap U_2)$, ${\mathcal{C}}^k$-diffeo.
    • intr.ch ${\mathcal{C}}^k$-compatible if such a mapping ex.: it is possible to der. on $M$ intrisically
  • a top.sp. $X$ is $m$-dim. loc.euc. if locally isom, to ${\mathbb{R}}^m$, i.e. each open ball is hom. to a subset of ${\mathbb{R}}^m$.
• Furhter Structure
  • an Atlas for $X$ loc.euc. is $\mathcal{A}\subseteq map(U_i,k_i{)}_{i\in I}$ s.t. $X={\bigcup }_{i\in I}{U}_i$.
    • $\mathcal{A}$ is ${\mathcal{C}}^k$-diff. (or ${\mathcal{C}}^k$-atl.) if its charts are pairw. ${\mathcal{C}}^k$-compatible.
    • each $\mathcal{A}$ ${\mathcal{C}}^k$-atl. has a unique max.${\mathcal{C}}^k$-atl. ${\mathcal{A}}^{\prime }$ s.t. $\mathcal{A}\subseteq {\mathcal{A}}^{\prime }$.
  • $F:X\to Y$ cont. is said diff. if for $k,k^{\prime }$ charts on $p$, $k^{-1}\circ F\circ k^{\prime }:k(X)\to k^{\prime }(X)$ is diff.
  • a $m$-top.man. is $m$-dim. loc.euc. Haus.sp. with count. basis
  • a $m$-dim ${\mathcal{C}}^k$-diff.man. is a $m$-dim.top.man. with ${\mathcal{C}}^k$-diff.struct.
    • a ${\mathcal{C}}^k$-diff.struc. on a loc.euc.sp. is a max.${\mathcal{C}}^k$-atl.
  • for $N^n$ a ${\mathcal{C}}^k$-mnf, $M^m\subseteq N^n$ is ${\mathcal{C}}^k$-submnf. of $N$ if for any $p\in M$ ex. $k_p:U\to V$ ($U\subseteq N$, $V\subseteq {\mathbb{R}}^n$) s.t. $k_p(M\cap U)=({\mathbb{R}}^m\times 0)\cap V$ an $m$-dim. ${\mathcal{C}}^k$-submnf. of ${\mathbb{R}}^m$.
    • ${\forall }_{p\in M}{\exists }_{k_p\in ch(U,k_p)}p\in dom(k_p)\wedge k(M\cap U)=({\mathbb{R}}^m\times 0)\cap V$ (Bright)

III.2 Manifolds with Boundaries

• $M\subseteq \Omega$ man. for $\Omega$ op., then $M$ has ${\mathcal{C}}^k$-bound. $\iff$ ${\forall }_{p\in \partial \Omega }{\exists }_{k\in ch(U,k)}K(\Omega \cap U)=H^m\cap k(U)$ Q
  • $f:U\to {\mathbb{R}}^m\in {\mathcal{C}}^{\infty }$, $U\subseteq {\mathbb{R}}^n$ Q (Regular Value Theorem) Bright

III.3

III.4 1-from and Cotangentbundle

• $M^m$ a ${\mathcal{C}}^k$-mnf, def. $df{|}_U={\sum }_{i=1}^m(\frac{\partial (f\circ x^{-1})}{\partial x_i}\circ x)d{x}_i=:{\sum }_{i=1}^m\frac{\partial f}{\partial x_i}d{x}_i$ (func. on $U\subset M$)
• $T_p^{\ast }M:=(T_{p}M{)}^{\ast }$, i.e. cotang.sp. is dualsp. of tang.sp.
• $T^{\ast }U={⨆}_{p\in U}{T}_p^{\ast }M$, $T^{\ast }x:T^{\ast }U\to x(U)\times {\mathbb{R}}^m,{\sum }_{i=1}^m{a}_i(d{x}_i{)}_p\mapsto (x(p),{\sum }_{i=1}^m{a}_i{e}_i)$, $a={\sum }_{i=1}^m{a}_i{e}_i$
• $d\widetilde{x_j}={\sum }_{i=1}^m(\frac{\partial (\widetilde{x_j}\circ x^{-1})}{\partial x_i}\circ x)d{x}_i=:{\sum }_{i=1}^m\frac{\partial \widetilde{x_j}}{\partial x_i}d{x}_i$ (change of basis)
• ${\sum }_{j=1}^m\widetilde{a_j}d\widetilde{x_j}={\sum }_{i=1}^m({\sum }_{j=1}^m\widetilde{a_j}(\frac{\partial {\tilde{x}}_j}{\partial x_i}\circ x))d{x}_i$,
  • loc.transf.map. $\tilde{x}(U\cap \tilde{U})\times {\mathbb{R}}^m\to x(U\cap \tilde{U})\times {\mathbb{R}}^m,(\tilde{x},\tilde{a})\mapsto (x(\tilde{x}),{\sum }_{i=1}^m({\sum }_{j=1}^m\widetilde{a_j}\frac{\partial \widetilde{x_j}}{\partial x_i}(x(\tilde{x})))e_i)$
  • and $(x,a)\mapsto (\tilde{x}(x),{\sum }_{j=1}^m({\sum }_{i=1}^{m}a\frac{\partial x_i}{\partial \widetilde{x_j}}(\tilde{x}(x)))e_j)$
    • $\tilde{a}={\sum }_{i=1}^m\widetilde{a_i}{e}_i$ and $a={\sum }_{i=1}^m{a}_i{e}_i$
• so $T^{\ast }M$ has the inducted strc. $2m$-dim. ${\mathcal{C}}^k$-mnf. with boundary.
• $1$-form $\alpha$ on $M$ is $l\le k-1$ cont.diff.func. $\alpha :M\to T^{\ast }M$ on $U$ iff. for all coeff. $\alpha {|}_U={\sum }_{i=1}^m{a}_{i}d{x}_i$
• for $F:M^m\to N^n$ diff. $d{F}_p:T_{p}M\to T_{F(p)}N$, then $d{F}_p^{\ast }:T_{F(p)}^{\ast }N\to T_p^{\ast }M,\lambda \mapsto \lambda \circ d{F}_p$ (pullback) and for $\alpha$ 1-from on $N$, $F^{\ast }\alpha$ 1-from on $M$ for $(F^{\ast }\alpha {)}_p:=d{F}_p^{\ast }({\alpha }_{F(p)})={\alpha }_{F(p)}\circ d{F}_p$.
• $F^{\ast }d{y}_j={\sum }_{i=1}^m((F^{\ast }d{y}_j)(\frac{\partial }{\partial x_i}))d{x}_i=:{\sum }_{i=1}^m(d{y}_j(d{F}_p\frac{\partial }{\partial x_i}))=:{\sum }_{i=1}^m(d(y_j\circ F\circ x^{-1})(e_i))d{x}_i$
• $d(y_j\circ F\circ x^{-1})(e_i)=\frac{\partial y_j}{\partial x_i}\circ x$ (also called $y(x)$) $\Rightarrow F^{\ast }d{y}_j={\sum }_{i=1}^m(\frac{\partial y_j}{\partial x_i}\circ x)d{x}_i$

III.5 Higher Dfferential Forms and External Derivatives

• For $M^m$ a ${\mathcal{C}}^l$-mnf- with bound., ${\Lambda }^{k}TM:={⨆}_{p\in M}{\Lambda }^k{T}_{p}M:={\Lambda }_k{T}_p^{\ast }M\cong Al{t}_k(T_{p}M)$ (Vectorspace of the alternating $k$-multilinearforms on $T_{p}M$)
  • ${\Lambda }^{\ast }TM:={⨆}_{p\in M}{\Lambda }^{\ast }{T}_{p}M={⨁}_{k=0}^m{\Lambda }^k{T}_{p}M$,
  • ${\pi }_{{\Lambda }^{k}TM}:{\Lambda }^{k}TM\to M$ and ${\pi }_{{\Lambda }^{\ast }TM}{\Lambda }^{\ast }TM\to M$
  • ${\Lambda }^{0}TM=\mathbb{R}$, ${\Lambda }^{1}TM=T^{\ast }M$, ${\Lambda }^{m}TM$ is determinant-lines-bundle
• a $k$-form (or diff.form of deg. $k$) on $M$ is an inters of the bundels ${\pi }_{{\Lambda }^{k}TM}$, ${\alpha }_M\to {\Lambda }^{k}TM$ s.t. ${\pi }_{{\Lambda }^{k}TM}\circ \alpha =i{d}_{{\Lambda }^{k}TM}\iff \alpha (p)\in {\Lambda }^k{T}_{p}M\iff \alpha (p)\in Al{t}_k(T_{p}M)$.
  • a mixed $k$-form is on $M$ is an inters. of bundles ${\pi }_{{\Lambda }^{\ast }TM}$ s.t. … (the same)
  • algebraic operations are pointwise $(\alpha +\beta {)}_p={\alpha }_p+{\beta }_p$, …
  • … Q
• $\alpha|U = \sum{1 \le i_1 < … < i_k \le m} a_{i_1, …, i_k} dx_{} idk
• for $F:M\to N$, $d{F}_p:T_p\to T_{F(p)}N$, then $d{F}_p^{\ast }:{\Lambda }^k{T}_{F(p)}N\to {\Lambda }^k{T}_{p}M$ is lin. and $\alpha$ $k$-form on $N$, then (pulled back $k$-from on M) $F^{\ast }\alpha$ is $(F^{\ast }\alpha )(v_1,...,v_k)={\alpha }_{F(p)}(d{F}_p(v_1),...,d{F}_p(v_k))$ for $v_1,...,v_k\in T_{p}M$
  • $F^{\ast }(\alpha \wedge \beta )=(F^{\ast }\alpha )\wedge (F^{\ast }\beta )$
  • $F^{\ast }(d{y}_{j_1}\wedge ...\wedge d{y}_{j_k})=(F^{\ast }d{y}_{j_1})\wedge ...\wedge (F^{\ast }d{y}_{j_k})={\sum }_{i_1=1}^m(\frac{\partial y_{j_1}}{\partial x_{i_1}}\circ x)d{x}_{i_1}=$ ${\sum }_{\le i_1,...,i_k\le m}({\prod }_{l=1}^k(\frac{\partial y_{j_l}}{\partial x_{i_l}}\circ x))d{x}_{i_1}\wedge ...\wedge d{x}_{i_k}=$ ${\sum }_{1\le i_1<...<i_k\le m}(det(\frac{\partial y_{j_r}}{\partial x_{i_s}})\circ x)d{x}_{i_1}\wedge ...\wedge d{x}_{i_k}$
    • for $k=m=n$ F^* dy_1 \land … \land dy_m = det (\frac{\partialy_r}{\partial x_s} \circ x)$ add substripts
  • synt. all the same for ${\Lambda }^{\ast }$ instead of ${\Lambda }^k$.
• External Derivative
  • the derivative $f\mapsto df$ from diff.func. to $1$-forms is a diff.operator of first order (ext. der.)
    • $d(fg)=(df)g+f(dg)$
    • ex.uniq. ext. of the total differential on func. to a fam of $\mathbb{R}$-ln. operators s.t.
      1. $d(\alpha \wedge \beta )=(d\alpha )\wedge \beta +(-1{)}^k\alpha \wedge d\beta$ for $\alpha$ diff.$k$-form, $\beta$ diff.$l$-fom (product rule)
      2. $d^2=0$ on $2$-times diff. forms (if $M$ is ${\mathcal{C}}^x$ for $x\ge 2$, else on func.) (comp. prop.)
    • for $p\in U$, $U\subseteq M$ $\Rightarrow$ ex. $\psi \in {\mathcal{C}}^2$ s.t. $supp(\psi )\subseteq U$, $supp(\psi )$ comp. (supp), $\psi \equiv 1$ cl. to $p$
  • for op. $d$ s.t. 1. & 2., $\alpha$ diff.form on $M$, $p\in M$ then ${\forall }_{U\subseteq M}p\in U\to {\exists }_{\psi \in {\mathcal{C}}^2(M)}\psi (U)\equiv 1$, $d(\psi \alpha )=(d\psi )\wedge \alpha +\psi d\alpha$, around $p$ $d(\psi \alpha )=d\alpha$, i.e. $(d\alpha {)}_p$ depends on $\alpha (p)$ (locality)
    • $(d_U{\alpha }_U{)}_p:=(d\alpha {)}_p$
    • for $\alpha ={\sum }_{i_1<...<i_k}{a}_{i_1,...,i_k}d{x}_{i_1}\wedge ...\wedge d{x}_{i_k}$ diff. on $U$ then by 1. & 2. we must: $d_U({\sum }_{i_1<...<i_k}{a}_{i_1,...,i_k}d{x}_{i_1}\wedge ...\wedge d{x}_{i_k}):={\sum }_{i_1<...<i_k}d{a}_{i_1,...,i_k}\wedge d{x}_{i_1}\wedge ...\wedge d{x}_{i_k}$
      1. for $\alpha =ad{x}_{i_1}\wedge ...\wedge d{x}_{i_k}$, $\beta =bd{x}_{j_1}\wedge ...\wedge d{x}_{j_l}$, (prod. rule) $d_U(\alpha \wedge \beta )=d_U(abd{x}_{i_1}\wedge ...\wedge d{x}_{i_k}\wedge d{x}_{j_1}\wedge ...\wedge d{x}_{j_l})=d(ab)\wedge d{x}_{i_1}\wedge ...=(da)b+a(db)\wedge d{x}_{i_1}\wedge ...$
      2. $d_U^2=d_U(df)=d_U({\sum }_i\frac{\partial f}{\partial x_i}d{x}_i)$ for $\frac{\partial f}{\partial x_i}=\frac{\partial (f\circ x^{-1})}{\partial x_i}\circ x$, then (complex property) $={\sum }_{i=1}^m(d\frac{\partial f}{\partial x_i})\wedge d{x}_i={\sum }_{i,j}\frac{{\partial }^{2}f}{\partial x_j\partial x_i}d{x}_j\wedge d{x}_i={\sum }_{j<i}(\frac{{\partial }^{2}f}{\partial x_j\partial x_i}-\frac{{\partial }^{2}f}{\partial x_i\partial x_j})d{x}_j\wedge d{x}_i=0$
    • …
• Recap Linear Algebra
  • $V$ is $m$-dim. $\mathbb{R}$-VS., the basis of $V$ are in $V\times ...\times V$ is open and dense ? has two components
    • you cannot get from a component to the other without going out of the basis.
    • The two basis are called positive and negative.
    • Boh

Manifolds: by Youtube: Bright Side

• 1. Introduction and Topology
  • for a met.sp. $(X,d)$ we can def. op. sets $A\subseteq X$, then:
  • $(X,\mathcal{T})$ is a top.sp. for $\mathcal{T}\subseteq \mathcal{P}(X)$ if (i) $\varnothing ,x\in \mathcal{T}$, (ii) $A,B\in \mathcal{T}\Rightarrow A\cap B\in \mathcal{T}$, (iii) $(A_i{)}_{i\in I}\subset \mathcal{T}\Rightarrow {\bigcup }_{i\in I}{A}_i\in \mathcal{T}$. each elem. in $\mathcal{T}$ are called open sets.
• 2. Interior, Exterior, Boundary, Closure
  • Analysis II (Lecture)
• 3. Hausdorff Spaces
  • $(a_n{)}_{n\in \mathbb{N}}\subseteq X$, for $a_n\to a:={\forall }_{U\in \mathcal{T}}a\in U\to {\exists }_{N\in \mathbb{N}}{\forall }_{n\ge N}{a}_n\in U$
  • a top.sp. is Hausd. if ${\forall }_{x,y\in X}x\ne y\to {\exists }_{U_x,U_y\in \mathcal{T}}x\in U_x\wedge y\in U_y\wedge U_x\cap U_y=\varnothing$

• 4. Quotient Space

• 5. Projective Space
  • $S^n:=\{x\in {\mathbb{R}}^{n+1}:||x||=1\}$
  • a proj.sp. $P^n(\mathbb{R}):=S^n/\sim$ for $x\sim y:=x=y\vee x=-y$
• 6. Second-Countable Space
  • generators of $\mathcal{T}$ is called a basis
  • $\mathcal{B}\subseteq \mathcal{T}$ is a basis if: ${\forall }_{U\in \mathcal{T}}{\exists }_{(A_i{)}_{i\in I}\subseteq \mathcal{B}}{\bigcup }_{i\in I}{A}_i=U$
  • $(X,\mathcal{T})$ is second-countable if there is a basis $\mathcal{B}$ s.t. $|\mathcal{B}|<{\aleph }_0$.
• 7. Continuity
  • $f:X\to Y$ is cont. if $f(U)$ op. $\Rightarrow U$ op.
  • $f$ homeom. if $f:X\to Y$ bij. cont. and $f^{-1}$ cont.
  • $f$ cont. $\iff$ for $(x_n{)}_{n\in \mathbb{N}}$ s.t. $x_n\to x$ then $(f(x_n){)}_{n\in \mathbb{N}}\to f(x)$
    • this holds in met.sp. and second-countable spaces.
• 8. Compactness
  • Analysis II (Lecture)
• 9. Locally Euclidean Spaces
  • $(X,\mathcal{T})$ is a mnf. if (i) Hausd., (ii) second-count., (iii) locally euc. of dim. $n$
  • $(X,\mathcal{T})$ is loc.euc. if ${\forall }_{x\in X}{\exists }_{U\in \mathcal{T}}{\exists }_{U^{\prime }\subset {\mathbb{R}}^n}{\exists }_{h:U\to U^{\prime }}h$ is homeom.
  • a trans.map.
• 16. Smooth Maps
  • $f:M\to N$ for $M,N$, $f$ diff in $p$ $\iff k\circ f\circ h^{-1}$ diff. in $p$.
• 19. Tangent Space
  • For $M\subseteq {\mathbb{R}}^n$ $k$-dim. mnf, $h:U\to U^{\prime }$, a param. $\phi :{\mathbb{R}}^k\cap U^{\prime }\to M\cap U$
  • $T_p^{sub}M:=d{\phi }_{\tilde{p}}[{\mathbb{R}}^{\mathbb{k}}]=\{J_{\phi }({\phi }^{-1}(p))x|x\in {\mathbb{R}}^k\}\subseteq {\mathbb{R}}^n$
    • $\tilde{p}:={\phi }^{-1}(p)$
  • ex. for $f:{\mathbb{R}}^2\to \mathbb{R}$, $f\in {\mathcal{C}}^1({\mathbb{R}}^2)$, $M=G_f:=\{(\begin{array}{c}x\\ y\\ f(x,y)\end{array})|(x,y)\in {\mathbb{R}}^2\}$, a param. $\phi :{\mathbb{R}}^2\to M,(\begin{array}{c}x\\ y\end{array})\mapsto (\begin{array}{c}x\\ y\\ f(x,y)\end{array})$, $T_p^{sub}M$ is the span of the columns of $J_{\phi }(x,y)$
• 20. Tangent Curves
  • $\gamma :\mathbb{R}\to M$ is a curve
  • $T_{p}M=\{{\gamma }^{\prime }(0)|\gamma :(-\epsilon ,\epsilon )\to M$ diff. $\wedge \gamma (0)=p\}$
• 21. Tangent Space (Definition via tangent courves)
  • Here we don’t require $M\subseteq {\mathbb{R}}^n$ as did in 19.
  • $C_p(M):=\{\gamma :(-\epsilon ,\epsilon )\to M|\gamma$ diff. $\wedge \gamma (0)=p\}$
  • $\gamma \sim \alpha :=(h\circ \gamma {)}^{\prime }(0)=(h\circ \alpha {)}^{\prime }(0)$
  • $[\gamma {]}_{\sim }:=\{\alpha |\gamma \sim \alpha \}$
  • $T_p(M):=C_p(M)/\sim$
    • $v+w:=h_{\ast }^{-1}(h_{\ast }(v)+h_{\ast }(w))$ for $h_{\ast }:[\gamma {]}_{\sim }\mapsto (h\circ \gamma {)}^{\prime }(0)$
    • $\lambda \cdot v:=h_{\ast }^{-1}(\lambda \cdot h_{\ast }(v))$
  • $T_p^{sub}M\to T_{p}M,{\gamma }^{\prime }(0)\mapsto [\gamma {]}_{\sim }$ nat. bij.
• 22. Coordinate Basis
  • $U\subseteq M$, $h:U\to U^{\prime }$, $\phi :=h^{-1}$, then $h_{\ast }:T_{p}M\to \mathbb{R},[\gamma ]\mapsto (h\circ \gamma {)}^{\prime }(0)$ lin. bij. ($\phi :=h_{\ast }^{-1}$)
  • a coor.bas. on $M$ resp. to $h$ chart, $({\partial }_j{)}_{j\in [1,...,n]}:=({\phi }_{\ast }(e_j){)}_{j\in [1,...,n]}$ is a basis of $T_{p}M$
    • ${\partial }_j=\frac{\partial \phi }{\partial x_j}$ this is why we use the notation ${\partial }_j$.
  • $T_{p}M{\cong }^{{\phi }_{\ast }}{\mathbb{R}}^n{\cong }^{J_{\phi }(\tilde{p})}{T}_p^{sub}M$
  • for $f:M\to N$ sm., $d{f}_p:T_{p}M\to T_{p}N$
• 23. Differential (Definition)
  • a tangent bundle $TM:={⨆}_{p\in M}{T}_{p}M$
    • $dim(TM)=2\cdot dim(M)$
  • for $f:M\to N$, $d{f}_p:T_{p}M\to T_{f(p)}N,[\gamma ]\mapsto [f\circ \gamma ]$
    • $df:p\mapsto d{f}_p$
  • $M,N\subseteq {\mathbb{R}}^n$ sm. $d{f}_p:[\gamma ]\mapsto [f\circ \gamma ]=(f\circ \gamma {)}^{\prime }(0)=J_f(\gamma (0)){\gamma }^{\prime }(0)$ i.e. der. of $f$ along $[\gamma ]$ at $p$.
• 24. Differential in Local Charts
  • $\tilde{f}:=k\circ f\circ h^{-1}$, $f=k^{-1}\circ \tilde{f}\circ h$, df = dk^{-1} J_\tilde{f} dh.
    • $d{k}_{f(p)}(d{f}_p([\gamma ]))=J_{\tilde{f}}(h(p)(h\circ \gamma {)}^{\prime }(0)=J_{\tilde{f}}[h\circ \gamma ]=J_{\tilde{f}}dh([\gamma ])$
• 25. Differential (Example)
  • ${\partial }_j={\phi }_{\ast }(e_j)=d_{{\phi }_{h(p)}}(e_j)$
  • for $f:M\to \mathbb{R}$, ${\partial }_{j}f(p):=d{f}_p({\partial }_j)=\frac{\partial (f\circ \phi )}{\partial x_j}(h(p))$
  • $[f\circ \phi \circ \tilde{\gamma }]=(f\circ \phi \tilde{\gamma }{)}^{\prime }(0)=J_{f\circ \phi }(h(p)){\tilde{\gamma }}^{\prime }(0)=\frac{\partial (f\circ \phi )}{\partial x_j}(h(p))$
    • $\tilde{\gamma }(t)=h(p)+t\cdot e_j$

Verschiedenes

• Dichte (B. 10): for $f\in {\mathcal{M}}_{[0,\infty ]}(X,\mathcal{A})$, then $f\mu :\mathcal{A}\to [0,\infty ],A\mapsto {\int }_{A}fd\mu$
  • (Wiki) if $X_1=f^{-1}(X_2)$, then ${\int }_{X_2}gd(f_{\ast }\mu )={\int }_{X_1}g\circ fd\mu$.
  • Existence