View source for Vitali–Hahn–Saks theorem

In [[mathematics]], the '''Vitali–Hahn–Saks theorem''', introduced by  {{harvs|txt|authorlink=Giuseppe Vitali|last=Vitali|year=1907}}, {{harvs|txt|authorlink=Hans Hahn (mathematician)|last=Hahn|year=1922}}, and {{harvs|txt|authorlink=Stanisław Saks|last=Saks|year=1933}}, proves that under some conditions a sequence of [[Measure_(mathematics)|measures]] converging point-wise does so uniformly and the limit is also a measure.

==Statement of the theorem==

If <math>(S,\mathcal{B},m)</math> is a [[measure space]] with <math>m(S)<\infty,</math> and a sequence <math>\lambda_n</math> of [[complex measure]]s. Assuming that each <math>\lambda_n</math> is [[Absolute_continuity#Absolute_continuity_of_measures|absolutely continuous]] with respect to <math>m,</math> and that for all <math>B\in\mathcal{B}</math> the finite limits exist <math>\lim_{n\to\infty}\lambda_n(B)=\lambda(B)</math>. Then the absolute continuity of the <math>\lambda_n</math> with respect to <math>m</math> is uniform in <math>n</math>, that is, <math>\lim_B m(B)=0</math> implies that <math>\lim_{B}\lambda_n(B)=0</math> uniformly in <math>n</math>. Also <math>\lambda</math> is countably additive on <math>\mathcal{B}</math>.

==Preliminaries==

Given a measure space <math>(S,\mathcal{B},m),</math> a distance can be constructed on <math>\mathcal{B}_0,</math> the set of measurable sets <math>B\in\mathcal{B}</math> with <math>m(B) < \infty.</math> This is done by defining 
:<math>d(B_1,B_2) = m(B_1\Delta B_2),</math> where <math>B_1\Delta B_2 = (B_1\setminus B_2) \cup (B_2\setminus B_1)</math> is the [[symmetric difference]] of the sets <math>B_1,B_2\in\mathcal{B}_0.</math>
This gives rise to a metric space <math>\tilde{\mathcal{B}_0}</math> by identifying two sets <math>B_1,B_2\in \mathcal{B}_0</math> when <math>m(B_1\Delta B_2)=0.</math> Thus a point <math>\overline{B}\in\tilde{\mathcal{B}_0}</math> with representative <math>B\in\mathcal{B}_0</math> is the set of all <math>B_1\in\mathcal{B}_0</math> such that <math>m(B\Delta B_1) = 0.</math>

'''Proposition:''' <math>\tilde{\mathcal{B}_0}</math> with the metric defined above is a [[complete metric space]].

''Proof:'' Let 
<math display=block>\chi_B(x)=\begin{cases}1,&x\in B\\0,&x\notin B\end{cases}</math>
Then
<math display=block>d(B_1,B_2)=\int_S|\chi_{B_1}(s)-\chi_{B_2}(x)|dm</math>
This means that the metric space <math>\tilde{\mathcal{B}_0}</math> can be identified with a subset of the [[Banach space]] <math>L^1(S,\mathcal{B},m)</math>.

Let <math>B_n\in\mathcal{B}_0</math>, with
<math display=block>\lim_{n,k\to\infty}d(B_n,B_k)=\lim_{n,k\to\infty}\int_S|\chi_{B_n}(x)-\chi_{B_k}(x)|dm=0</math>
Then we can choose a sub-sequence <math>\chi_{B_{n'}}</math> such that <math>\lim_{n'\to\infty}\chi_{B_{n'}}(x)=\chi(x)</math> exists [[almost everywhere]] and <math>\lim_{n'\to\infty}\int_S|\chi(x)-\chi_{B_{n'}(x)}|dm=0</math>. It follows that <math>\chi=\chi_{B_{\infty}}</math> for some <math>B_{\infty}\in\mathcal{B}_0</math> (furthermore <math>\chi (x) = 1</math> [[if and only if]] <math>\chi_{B_{n'}} (x) = 1</math> for <math>n'</math> large enough, then we have that <math>B_{\infty} = \liminf_{n'\to\infty}B_{n'} = {\bigcup_{n'=1}^\infty}\left({\bigcap_{m=n'}^\infty}B_m\right) </math> the [[limit inferior]] of the sequence) and hence <math>\lim_{n\to\infty}d(B_\infty,B_n)=0.</math> Therefore, <math>\tilde{\mathcal{B}_0}</math> is complete.

==Proof of Vitali-Hahn-Saks theorem==

Each <math>\lambda_n</math> defines a function <math>\overline{\lambda}_n(\overline{B})</math> on <math>\tilde{\mathcal{B}}</math> by taking <math>\overline{\lambda}_n(\overline{B})=\lambda_n(B)</math>. This function is well defined, this is it is independent on the representative <math>B</math> of the class <math>\overline{B}</math> due to the absolute continuity of <math>\lambda_n</math> with respect to <math>m</math>. Moreover <math>\overline{\lambda}_n</math> is continuous. 

For every <math>\epsilon>0</math> the set
<math display=block>F_{k,\epsilon}=\{\overline{B}\in\tilde{\mathcal{B}}:\ \sup_{n\geq1}|\overline{\lambda}_k(\overline{B})-\overline{\lambda}_{k+n}(\overline{B})|\leq\epsilon\}</math>
is closed in <math>\tilde{\mathcal{B}}</math>, and by the hypothesis <math>\lim_{n\to\infty}\lambda_n(B)=\lambda(B)</math> we have that 
<math display=block>\tilde{\mathcal{B}}=\bigcup_{k=1}^{\infty}F_{k,\epsilon}</math>
By [[Baire category theorem]] at least one <math>F_{k_0,\epsilon}</math> must contain a non-empty open set of <math>\tilde{\mathcal{B}}</math>. This means that there is <math>B_0\in \mathcal{B}</math> and a <math>\delta>0</math> such that 
<math display=block>d(B,B_0)<\delta \Rightarrow\sup_{n\geq1}|\overline{\lambda}_{k_0}(\overline{B})-\overline{\lambda}_{k_0+n}(\overline{B})|\leq\epsilon.</math>
On the other hand, any <math>B\in\mathcal{B}</math> with <math>m(B)\leq\delta</math> can be represented as <math>B=B_1\setminus B_2</math> with <math>d(B_1,B_0)\leq\delta</math> and <math>d(B_2,B_0)\leq \delta</math>. This can be done, for example by taking <math>B_1=B\cup B_0</math> and <math>B_2=B_0\setminus(B\cap B_0)</math>. Thus, if <math>m(B)\leq\delta</math> and <math>k\geq k_0</math> then
<math display=block>\begin{align}|\lambda_k(B)|&\leq|\lambda_{k_0}(B)|+|\lambda_{k_0}(B)-\lambda_k(B)|\\&\leq|\lambda_{k_0}(B)|+|\lambda_{k_0}(B_1)-\lambda_k(B_1)|+|\lambda_{k_0}(B_2)-\lambda_k(B_2)|\\&\leq|\lambda_{k_0}(B)|+2\epsilon\end{align}</math>
Therefore, by the absolute continuity of <math>\lambda_{k_0}</math> with respect to <math>m</math>, and since <math>\epsilon</math> is arbitrary, we get that <math>m(B)\to0</math> implies <math>\lambda_n(B) \to 0</math> uniformly in <math>n.</math> In particular, <math>m(B) \to 0</math> implies <math>\lambda(B) \to 0.</math> 

By the additivity of the limit it follows that <math>\lambda</math> is [[Sigma_additivity#Additive_(or_finitely_additive)_set_functions|finitely-additive]]. Then, since <math>\lim_{m(B) \to 0}\lambda(B) = 0</math> it follows that <math>\lambda</math> is actually countably additive.

==References==

{{reflist}}

*{{Citation | last1=Hahn | first1=H. | author1-link=Hans Hahn (mathematician) | title=Über Folgen linearer Operationen | language=German | year=1922 | journal=Monatsh. Math. | volume=32 | pages=3–88 | doi=10.1007/bf01696876| url=https://zenodo.org/record/1428338 }}
*{{Citation | last1=Saks | first1=Stanislaw | author1-link=Stanislaw Saks | title=Addition to the Note on Some Functionals | year=1933 | journal=[[Transactions of the American Mathematical Society]] | volume=35 | issue=4 | pages= 965–970 | doi=10.2307/1989603 | jstor=1989603 }}
*{{Citation | last1=Vitali | first1=G. |author1-link=Giuseppe Vitali| title=Sull' integrazione per serie | language=Italian | doi=10.1007/BF03013514 | year=1907 | journal=[[Rendiconti del Circolo Matematico di Palermo]] | volume=23 | pages=137–155| url=https://zenodo.org/record/2082522 }}
*{{Citation | last1=Yosida| first1=K. | title=Functional Analysis | publisher=Springer | year=1971 |isbn=0-387-05506-1 | pages=70–71}}

{{Measure theory}}

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[[Category:Theorems in measure theory]]