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There are two important theorems which make statements about convergence types of measurable maps. They are important to understand the behavior of sequence of measurable maps and to understand the importance of uniform convergence. | |||
{{proofcard|Theorem (Egorov)|thm-1|Let <math>(E,\A,\mu)</math> be a measure space. Let <math>f_k:E\longrightarrow \bar \R</math> be measurable for all <math>k\in\mathbb{N}</math> and <math>f:E\longrightarrow\bar \R</math> be measurable and <math>\mu</math>-a.e. finite. Moreover <math>f_k(x)\xrightarrow{k\to\infty} f(x)</math> <math>\mu</math>-a.e. for <math>x\in E</math>. Then for all <math>\delta > 0</math> there exists <math>F\subset E</math>, with <math>F</math> compact and <math>\mu(E\setminus F) < \delta</math> and | |||
<math display="block"> | |||
\sup_{x\in F}\vert f_k(x)-f(x)\vert\xrightarrow{k\to\infty} 0, | |||
</math> | |||
i.e. <math>(f_k)_{k\in\mathbb{N}}</math> converges uniformly to <math>f</math> in <math>F</math>. | |||
|Let <math>\delta > 0</math>. For <math>i,j\in\mathbb{N}</math> set | |||
<math display="block"> | |||
C_{i,j}:=\bigcup_{k=j}^{\infty}\{x\in E\mid \vert f_k(x)-f(x)\vert > 2^{-1}\}. | |||
</math> | |||
<math>C_{i,j}</math> is <math>\mu</math>-measurable, because <math>f</math> and <math>f_k</math> are <math>\mu</math>-measurable and <math>C_{i,(j+1)}\subset C_{i,j}</math>, <math>\forall i,j</math>. | |||
We also know that <math>f_k(x)\xrightarrow{k\to\infty}f(x)</math> for <math>\mu</math>-a.e. <math>x\in E</math> and since <math>\mu(E) < \infty</math> it follows that for all <math>i\in\N</math> | |||
<math display="block"> | |||
\lim_{j\to\infty}\mu(C_{i,j})=\mu\left(\bigcup_{j=1}^\infty C_{i,j}\right)=0. | |||
</math> | |||
So for every <math>i</math> there exists a <math>N(i)\in\N</math> with | |||
<math display="block"> | |||
\mu(C_{i,N(i)}) < \delta\cdot 2^{-i-1}. | |||
</math> | |||
Now set <math>A=E\setminus \bigcup_{i=1}^\infty C_{i,N(i)}</math>. Then | |||
<math display="block"> | |||
\mu(E\setminus A)\leq \sum_{i=1}^{\infty}\mu(C_{i,N(i)}) < \delta/2, | |||
</math> | |||
and for all <math>i\in\N</math> and <math>k\geq N(i)</math> | |||
<math display="block"> | |||
\sup_{x\in A}\vert f_k(x)-f(x)\vert \leq 2^{-i}. | |||
</math> | |||
Choose a <math>F\subset A</math>, where <math>F</math> is compact with <math>\mu(A\setminus F) < \delta/2</math>. Hence we have | |||
<math display="block"> | |||
\mu(E\setminus F)\leq \mu(E\setminus A)+\mu(A\setminus F) < \delta. | |||
</math>}} | |||
{{proofcard|Theorem (Lusin)|thm-2|Let <math>(E,\A,\mu)</math> be a measure space. Let <math>f:E\longrightarrow\bar \R</math> be measurable and <math>\mu</math>-a.e. finite. Then for all <math>\delta > 0</math> there exists <math>F\subset E</math>, <math>F</math> compact with <math>\mu(E\setminus F) < \delta</math> and <math>f\mid_F:F\longrightarrow \R</math> is continuous. | |||
|We split the proof onto two parts. | |||
<ul style{{=}}"list-style-type:lower-roman"><li>We are going to show this theorem for step functions of the form | |||
<math display="block"> | |||
g=\sum_{i=1}^Ib_i\one_{B_i}, | |||
</math> | |||
where we set <math>E=\bigsqcup_{i=1}^{I}B_i</math> with <math>B_i\cap B_j=\varnothing</math> for <math>i\not=j</math>. For <math>\delta > 0</math> choose <math>F_i\subset B_i</math> compact with | |||
<math display="block"> | |||
\mu(B_i\setminus F_i) < \delta\cdot 2^{-i},1\leq i\leq I | |||
</math> | |||
Since the sets <math>B_i</math> are disjoint, it follows that the sets <math>F_i</math> are also disjoint, because of the fact that they are also compact it follows that <math>d(F_i,F_j) > 0</math> for <math>i\not=j</math>. Therefore we notice that <math>g</math> is locally constant, i.e. continuous on <math>F:=\bigcup_{i=1}^IF_i\subset E.</math> Moreover <math>F\subset E</math> and | |||
<math display="block"> | |||
\mu(E\setminus F)=\mu\left(\bigcup_{i=1}^I(B_i\setminus F_i)\right)\leq \sum_{i=1}^I\mu(B_i\setminus F_i) < \delta. | |||
</math> | |||
</li> | |||
<li>Let <math>f_k:E\longrightarrow \R</math> be a step function with | |||
<math display="block"> | |||
f(x)=\lim_{k\to\infty}f_k(x),x\in E, | |||
</math> | |||
where | |||
<math display="block"> | |||
f_k=\sum_{j=1}^k\frac{1}{j}\one_{A_j}=\sum_{i=1}^{I_k}b_{ik}\one_{B_{ik}},k\in\N, | |||
</math> | |||
with <math>B_{ik}\cap B_{jk}=\varnothing</math> for <math>i\not=j</math> and <math>\bigsqcup_{i=1}^{I_k}B_{ik}=E</math> and with | |||
<math display="block"> | |||
b_{ik}=\sum_{B_{ik}\subset A_j}\frac{1}{j},1\leq i\leq I_k,k\in\N. | |||
</math> | |||
For <math>\delta > 0</math>, <math>g=f_k</math> choose compact sets <math>F_k\in E</math> as in part <math>(i)</math> with | |||
<math display="block"> | |||
\mu(E\setminus F_k) < \delta\cdot 2^{-k-1},f_k|_{F_k}:F_k\longrightarrow \R\text{continuous,}k\in\N. | |||
</math> | |||
Choose also <math>F_0\subset E</math> compact with | |||
<math display="block"> | |||
\mu(E\setminus F_0) < \delta/2,\sup_{x\in F_0}\vert f_k(x)-f(x)\vert\xrightarrow{k\to\infty} 0. | |||
</math> | |||
Finally let <math>F=\bigcap_{k=0}^\infty F_k\subset E</math> Note that <math>F</math> is compact with | |||
<math display="block"> | |||
\mu(E\setminus F)\leq \mu\left(\bigcup_{k=0}^\infty(E\setminus F_k)\right)\leq \sum_{k=0}^\infty\mu(E\setminus F_k) < \delta. | |||
</math> | |||
and because of the fact that <math>F\subset F_0</math> it follows that | |||
<math display="block"> | |||
\sup_{x\in F}\vert f_k(x)-f(x)\vert\xrightarrow{k\to\infty} 0. | |||
</math> | |||
The continuity of <math>f_k\mid_{F}</math>, <math>k\in\N</math>, gives us now the continuity of <math>f\mid_F:F\longrightarrow \R.</math> | |||
</li> | |||
</ul>}} | |||
==General references== | |||
{{cite arXiv|last=Moshayedi|first=Nima|year=2020|title=Lectures on Probability Theory|eprint=2010.16280|class=math.PR}} |
Latest revision as of 01:53, 8 May 2024
There are two important theorems which make statements about convergence types of measurable maps. They are important to understand the behavior of sequence of measurable maps and to understand the importance of uniform convergence.
Let [math](E,\A,\mu)[/math] be a measure space. Let [math]f_k:E\longrightarrow \bar \R[/math] be measurable for all [math]k\in\mathbb{N}[/math] and [math]f:E\longrightarrow\bar \R[/math] be measurable and [math]\mu[/math]-a.e. finite. Moreover [math]f_k(x)\xrightarrow{k\to\infty} f(x)[/math] [math]\mu[/math]-a.e. for [math]x\in E[/math]. Then for all [math]\delta \gt 0[/math] there exists [math]F\subset E[/math], with [math]F[/math] compact and [math]\mu(E\setminus F) \lt \delta[/math] and
i.e. [math](f_k)_{k\in\mathbb{N}}[/math] converges uniformly to [math]f[/math] in [math]F[/math].
Let [math]\delta \gt 0[/math]. For [math]i,j\in\mathbb{N}[/math] set
[math]C_{i,j}[/math] is [math]\mu[/math]-measurable, because [math]f[/math] and [math]f_k[/math] are [math]\mu[/math]-measurable and [math]C_{i,(j+1)}\subset C_{i,j}[/math], [math]\forall i,j[/math]. We also know that [math]f_k(x)\xrightarrow{k\to\infty}f(x)[/math] for [math]\mu[/math]-a.e. [math]x\in E[/math] and since [math]\mu(E) \lt \infty[/math] it follows that for all [math]i\in\N[/math]
So for every [math]i[/math] there exists a [math]N(i)\in\N[/math] with
Now set [math]A=E\setminus \bigcup_{i=1}^\infty C_{i,N(i)}[/math]. Then
and for all [math]i\in\N[/math] and [math]k\geq N(i)[/math]
Choose a [math]F\subset A[/math], where [math]F[/math] is compact with [math]\mu(A\setminus F) \lt \delta/2[/math]. Hence we have
Let [math](E,\A,\mu)[/math] be a measure space. Let [math]f:E\longrightarrow\bar \R[/math] be measurable and [math]\mu[/math]-a.e. finite. Then for all [math]\delta \gt 0[/math] there exists [math]F\subset E[/math], [math]F[/math] compact with [math]\mu(E\setminus F) \lt \delta[/math] and [math]f\mid_F:F\longrightarrow \R[/math] is continuous.
We split the proof onto two parts.
- We are going to show this theorem for step functions of the form
[[math]] g=\sum_{i=1}^Ib_i\one_{B_i}, [[/math]]where we set [math]E=\bigsqcup_{i=1}^{I}B_i[/math] with [math]B_i\cap B_j=\varnothing[/math] for [math]i\not=j[/math]. For [math]\delta \gt 0[/math] choose [math]F_i\subset B_i[/math] compact with[[math]] \mu(B_i\setminus F_i) \lt \delta\cdot 2^{-i},1\leq i\leq I [[/math]]Since the sets [math]B_i[/math] are disjoint, it follows that the sets [math]F_i[/math] are also disjoint, because of the fact that they are also compact it follows that [math]d(F_i,F_j) \gt 0[/math] for [math]i\not=j[/math]. Therefore we notice that [math]g[/math] is locally constant, i.e. continuous on [math]F:=\bigcup_{i=1}^IF_i\subset E.[/math] Moreover [math]F\subset E[/math] and[[math]] \mu(E\setminus F)=\mu\left(\bigcup_{i=1}^I(B_i\setminus F_i)\right)\leq \sum_{i=1}^I\mu(B_i\setminus F_i) \lt \delta. [[/math]]
- Let [math]f_k:E\longrightarrow \R[/math] be a step function with
[[math]] f(x)=\lim_{k\to\infty}f_k(x),x\in E, [[/math]]where[[math]] f_k=\sum_{j=1}^k\frac{1}{j}\one_{A_j}=\sum_{i=1}^{I_k}b_{ik}\one_{B_{ik}},k\in\N, [[/math]]with [math]B_{ik}\cap B_{jk}=\varnothing[/math] for [math]i\not=j[/math] and [math]\bigsqcup_{i=1}^{I_k}B_{ik}=E[/math] and with[[math]] b_{ik}=\sum_{B_{ik}\subset A_j}\frac{1}{j},1\leq i\leq I_k,k\in\N. [[/math]]For [math]\delta \gt 0[/math], [math]g=f_k[/math] choose compact sets [math]F_k\in E[/math] as in part [math](i)[/math] with[[math]] \mu(E\setminus F_k) \lt \delta\cdot 2^{-k-1},f_k|_{F_k}:F_k\longrightarrow \R\text{continuous,}k\in\N. [[/math]]Choose also [math]F_0\subset E[/math] compact with[[math]] \mu(E\setminus F_0) \lt \delta/2,\sup_{x\in F_0}\vert f_k(x)-f(x)\vert\xrightarrow{k\to\infty} 0. [[/math]]Finally let [math]F=\bigcap_{k=0}^\infty F_k\subset E[/math] Note that [math]F[/math] is compact with[[math]] \mu(E\setminus F)\leq \mu\left(\bigcup_{k=0}^\infty(E\setminus F_k)\right)\leq \sum_{k=0}^\infty\mu(E\setminus F_k) \lt \delta. [[/math]]and because of the fact that [math]F\subset F_0[/math] it follows that[[math]] \sup_{x\in F}\vert f_k(x)-f(x)\vert\xrightarrow{k\to\infty} 0. [[/math]]The continuity of [math]f_k\mid_{F}[/math], [math]k\in\N[/math], gives us now the continuity of [math]f\mid_F:F\longrightarrow \R.[/math]
General references
Moshayedi, Nima (2020). "Lectures on Probability Theory". arXiv:2010.16280 [math.PR].