In mathematics, the Vitali–Hahn–Saks theorem, introduced by Vitali (1907), Hahn (1922), and Saks (1933), proves that under some conditions a sequence of measures converging point-wise does so uniformly and the limit is also a measure.
Statement of the theorem
If \( {\displaystyle (S,{\mathcal {B}},m)} \) is a measure space with \( {\displaystyle m(S)<\infty } \), and a sequence λ n {\displaystyle \lambda _{n}} \lambda _{n} of complex measures. Assuming that each \( \lambda _{n} \) is absolutely continuous with respect to } m, and that a for all \( {\displaystyle B\in {\mathcal {B}}}\) the finite limits exist \( {\displaystyle \lim _{n\to \infty }\lambda _{n}(B)=\lambda (B)} \). Then the absolute continuity of the \( \lambda _{n} with respect to m {\displaystyle m} m is uniform in n, that is ,\( {\displaystyle \lim _{B}m(B)=0} \) implies that \({\displaystyle \lim _{B}\lambda _{n}(B)=0} \) uniformly in n. Also \( \lambda \) is countably additive on \({\mathcal {B}}. \)
Preliminaries
Given a measure space \( {\displaystyle (S,{\mathcal {B}},m)} \), a distance can be constructed on \( {\displaystyle {\mathcal {B}}_{0}} \), the set of measurable sets \( {\displaystyle B\in {\mathcal {B}}} \) with \( {\displaystyle m(B)<\infty } \). This is done by defining
\( {\displaystyle d(B_{1},B_{2})=m(B_{1}\Delta B_{2})} \), where \( {\displaystyle B_{1}\Delta B_{2}=(B_{1}\setminus B_{2})\cup (B_{2}\setminus B_{1})} \) is the symmetric difference of the sets \( {\displaystyle B_{1},B_{2}\in {\mathcal {B}}_{0}}. \)
This gives rise to a metric space \( {\displaystyle {\tilde {{\mathcal {B}}_{0}}}} \) by identifying two sets \( {\displaystyle B_{1},B_{2}\in {\mathcal {B}}_{0}} \) when \( {\displaystyle m(B_{1}\Delta B_{2})=0} \) . Thus a point \( {\displaystyle {\overline {B}}\in {\tilde {{\mathcal {B}}_{0}}}} \) with representative \( {\displaystyle B\in {\mathcal {B}}_{0}} \) is the set of all \( {\displaystyle B_{1}\in {\mathcal {B}}_{0}} \) such that \( {\displaystyle m(B\Delta B_{1})=0}. \)
Proposition: \( {\displaystyle {\tilde {{\mathcal {B}}_{0}}}} \) with the metric defined above is a complete metric space.
Proof: Let
\( {\displaystyle \chi _{B}(x)={\begin{cases}1,&x\in B\\0,&x\notin B\end{cases}}} \)
Then
\( {\displaystyle d(B_{1},B_{2})=\int _{S}|\chi _{B_{1}}(s)-\chi _{B_{2}}(x)|dm} \)
This means that the metric space \( {\displaystyle {\tilde {{\mathcal {B}}_{0}}}} \) can be identified with a subset of the Banach space \) {\displaystyle L^{1}(S,{\mathcal {B}},m)}. \)
Let \( {\displaystyle B_{n}\in {\mathcal {B}}_{0}} \) , with
\({\displaystyle \lim _{n,k\to \infty }d(B_{n},B_{k})=\lim _{n,k\to \infty }\int _{S}|\chi _{B_{n}}(x)-\chi _{B_{k}}(x)|dm=0} \)
Then we can choose a sub-sequence \( {\displaystyle \chi _{B_{n'}}} \) such that \( {\displaystyle \lim _{n'\to \infty }\chi _{B_{n'}}(x)=\chi (s)} \) exists almost everywhere and \( {\displaystyle \lim _{n'\to \infty }\int _{S}|\chi (x)-\chi _{B_{n'}(x)}|dm=0} \). It follows that \( {\displaystyle \chi =\chi _{B_{\infty }}} \) for some \( {\displaystyle B_{\infty }\in {\mathcal {B}}_{0}} \) and hence \( {\displaystyle \lim _{n\to \infty }d(B_{\infty },B_{n})=0} \). Therefore, \( {\displaystyle {\tilde {{\mathcal {B}}_{0}}}} \) is complete.
Proof of Vitali-Hahn-Saks theorem
Each \( \lambda _{n} \) defines a function \( {\displaystyle {\overline {\lambda }}_{n}({\overline {B}})} \) on\( {\displaystyle {\tilde {\mathcal {B}}}} \) by taking \( {\displaystyle {\overline {\lambda }}_{n}({\overline {B}})=\lambda _{n}(B)} \). This function is well defined, this is it is independent on the representative B of the class \( {\overline {B}} \) due to the absolute continuity of \( \lambda _{n} \) with respect to m. Moreover \( {\displaystyle {\overline {\lambda }}_{n}} \) is continuous.
For every \( \epsilon >0 \) the set
\( {\displaystyle F_{k,\epsilon }=\{{\overline {B}}\in {\tilde {\mathcal {B}}}:\ \sup _{n\geq 1}|{\overline {\lambda }}_{k}({\overline {B}})-{\overline {\lambda }}_{k+n}({\overline {B}})|\leq \epsilon \}} \)
is closed in \( {\displaystyle {\tilde {\mathcal {B}}}} \), and by the hypothesis lim \( {\displaystyle \lim _{n\to \infty }\lambda _{n}(B)=\lambda (B)} \) we have that
\( {\displaystyle {\tilde {\mathcal {B}}}=\bigcup _{k=1}^{\infty }F_{k,\epsilon }}
By Baire category theorem at least one \( {\displaystyle F_{k_{0},\epsilon }} \) must contain a non-empty open set of \( {\displaystyle {\tilde {\mathcal {B}}}} \). This means that there is \( {\displaystyle {\overline {B_{0}}}\in {\tilde {\mathcal {B}}}} \) and a \( \delta >0 \)such that
\( {\displaystyle d(B,B_{0})<\delta } \) implies \( {\displaystyle \sup _{n\geq 1}|{\overline {\lambda }}_{k_{0}}({\overline {B}})-{\overline {\lambda }}_{k_{0}+n}({\overline {B}})|\leq \epsilon } \)
On the other hand, any \( {\displaystyle B\in {\mathcal {B}}} \) with \( {\displaystyle m(B)\leq \delta } \) can be represented as \( {\displaystyle B=B_{1}\setminus B_{2}} \) with \( {\displaystyle d(B_{1},B_{0})\leq \delta } \) and \( {\displaystyle d(B_{2},B_{0})\leq \delta } \). This can be done, for example by taking \( {\displaystyle B_{1}=B\cup B_{0}} \) and \( {\displaystyle B_{2}=B_{0}\setminus (B\cap B_{0})} \). Thus, if \( {\displaystyle m(B)\leq \delta } \) and \( {\displaystyle k\geq k_{0}} \) then
\( {\displaystyle {\begin{aligned}|\lambda _{k}(B)|&\leq |\lambda _{k_{0}}(B)|+|\lambda _{k_{0}}(B)-\lambda _{k}(B)|\\&\leq |\lambda _{k_{0}}(B)|+|\lambda _{k_{0}}(B_{1})-\lambda _{k}(B_{1})|+|\lambda _{k_{0}}(B_{2})-\lambda _{k}(B_{2})|\\&\leq |\lambda _{k_{0}}(B)|+2\epsilon \end{aligned}}} \)
Therefore, by the absolute continuity of \( {\displaystyle \lambda _{k_{0}}} \) with respect to m, and since \( \epsilon \) is arbitrary, we get that \({\displaystyle m(B)\to 0} \) implies \( {\displaystyle \lambda _{n}(B)\to 0} \) uniformly in n. In particular, \( {\displaystyle m(B)\to 0} \) implies \( {\displaystyle \lambda (B)\to 0}. \)
By the additivity of the limit it follows that \( \lambda \) is finitely-additive. Then, since \( {\displaystyle \lim _{m(B)\to 0}\lambda (B)=0} \) it follows that \( \lambda \) is actually countably additive.
References
Hahn, H. (1922), "Über Folgen linearer Operationen", Monatsh. Math. (in German), 32: 3–88, doi:10.1007/bf01696876
Saks, Stanislaw (1933), "Addition to the Note on Some Functionals", Transactions of the American Mathematical Society, 35 (4): 965–970, doi:10.2307/1989603, JSTOR 1989603
Vitali, G. (1907), "Sull' integrazione per serie", Rendiconti del Circolo Matematico di Palermo (in Italian), 23: 137–155, doi:10.1007/BF03013514
Yosida, K. (1971), Functional Analysis, Springer, pp. 70–71, ISBN 0-387-05506-1
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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