Inmathematics, theVitali–Hahn–Saks theorem, introduced byVitali (1907),Hahn (1922), andSaks (1933), proves that under some conditions a sequence ofmeasures converging point-wise does so uniformly and the limit is also a measure.

If${\displaystyle (S,\mathcal{B},m)}$ is ameasure space with and a sequence${\displaystyle {\lambda }_n}$ ofcomplex measures. Assuming that each${\displaystyle {\lambda }_n}$ isabsolutely continuous with respect to${\displaystyle m,}$ and that for all${\displaystyle B\in \mathcal{B}}$ the finite limits exist${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{\lambda }_n(B)=\lambda (B)}$. Then the absolute continuity of the${\displaystyle {\lambda }_n}$ with respect to${\displaystyle m}$ is uniform in${\displaystyle n}$, that is,${\displaystyle \underset{B}{lim}m(B)=0}$ implies that${\displaystyle \underset{B}{lim}{\lambda }_n(B)=0}$ uniformly in${\displaystyle n}$. Also${\displaystyle \lambda }$ is countably additive on${\displaystyle \mathcal{B}}$.

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 This is done by defining

${\displaystyle d(B_1,B_2)=m(B_1\mathrm{\Delta }{B}_2),}$ where${\displaystyle B_1\mathrm{\Delta }{B}_2=(B_1\setminus B_2)\cup (B_2\setminus B_1)}$ is thesymmetric difference of the sets${\displaystyle B_1,B_2\in {\mathcal{B}}_0.}$

This gives rise to a metric space${\displaystyle \widetilde{{\mathcal{B}}_0}}$ by identifying two sets${\displaystyle B_1,B_2\in {\mathcal{B}}_0}$ when${\displaystyle m(B_1\mathrm{\Delta }{B}_2)=0.}$ Thus a point${\displaystyle \overline{B}\in \widetilde{{\mathcal{B}}_0}}$ with representative${\displaystyle B\in {\mathcal{B}}_0}$ is the set of all${\displaystyle B_1\in {\mathcal{B}}_0}$ such that${\displaystyle m(B\mathrm{\Delta }{B}_1)=0.}$

Proposition:${\displaystyle \widetilde{{\mathcal{B}}_0}}$ with the metric defined above is acomplete metric space.

Proof: LetThen$${\displaystyle d(B_1,B_2)={\int }_S|{\chi }_{B_1}(s)-{\chi }_{B_2}(x)|dm}$$This means that the metric space${\displaystyle \widetilde{{\mathcal{B}}_0}}$ can be identified with a subset of theBanach space${\displaystyle L^1(S,\mathcal{B},m)}$.

Let${\displaystyle B_n\in {\mathcal{B}}_0}$, with$${\displaystyle \underset{n,k\to \mathrm{\infty }}{lim}d(B_n,B_k)=\underset{n,k\to \mathrm{\infty }}{lim}{\int }_S|{\chi }_{B_n}(x)-{\chi }_{B_k}(x)|dm=0}$$Then we can choose a sub-sequence${\displaystyle {\chi }_{B_{n^{\prime }}}}$ such that${\displaystyle \underset{n^{\prime }\to \mathrm{\infty }}{lim}{\chi }_{B_{n^{\prime }}}(x)=\chi (x)}$ existsalmost everywhere and${\displaystyle \underset{n^{\prime }\to \mathrm{\infty }}{lim}{\int }_S|\chi (x)-{\chi }_{B_{n^{\prime }}(x)}|dm=0}$. It follows that${\displaystyle \chi ={\chi }_{B_{\mathrm{\infty }}}}$ for some${\displaystyle B_{\mathrm{\infty }}\in {\mathcal{B}}_0}$ (furthermore${\displaystyle \chi (x)=1}$if and only if${\displaystyle {\chi }_{B_{n^{\prime }}}(x)=1}$ for${\displaystyle n^{\prime }}$ large enough, then we have that${\displaystyle B_{\mathrm{\infty }}=\underset{n^{\prime }\to \mathrm{\infty }}{lim inf}{B}_{n^{\prime }}=\bigcup _{n^{\prime }=1}^{\mathrm{\infty }}\left(\bigcap _{m=n^{\prime }}^{\mathrm{\infty }}{B}_m\right)}$ thelimit inferior of the sequence) and hence${\displaystyle \underset{n\to \mathrm{\infty }}{lim}d(B_{\mathrm{\infty }},B_n)=0.}$ Therefore,${\displaystyle \widetilde{{\mathcal{B}}_0}}$ is complete.

Each${\displaystyle {\lambda }_n}$ 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${\displaystyle B}$ of the class${\displaystyle \overline{B}}$ due to the absolute continuity of${\displaystyle {\lambda }_n}$ with respect to${\displaystyle m}$. Moreover${\displaystyle {\overline{\lambda }}_n}$ is continuous.

For every${\displaystyle ϵ>0}$ the set$${\displaystyle F_{k,ϵ}=\{\overline{B}\in \tilde{\mathcal{B}}:\underset{n\ge 1}{sup}|{\overline{\lambda }}_k(\overline{B})-{\overline{\lambda }}_{k+n}(\overline{B})|\le ϵ\}}$$is closed in${\displaystyle \tilde{\mathcal{B}}}$, and by the hypothesis${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{\lambda }_n(B)=\lambda (B)}$ we have that$${\displaystyle \tilde{\mathcal{B}}=\bigcup _{k=1}^{\mathrm{\infty }}{F}_{k,ϵ}}$$ByBaire category theorem at least one${\displaystyle F_{k_0,ϵ}}$ must contain a non-empty open set of${\displaystyle \tilde{\mathcal{B}}}$. This means that there is${\displaystyle B_0\in \mathcal{B}}$ and a${\displaystyle \delta >0}$ such thatOn the other hand, any${\displaystyle B\in \mathcal{B}}$ with${\displaystyle m(B)\le \delta }$ can be represented as${\displaystyle B=B_1\setminus B_2}$ with${\displaystyle d(B_1,B_0)\le \delta }$ and${\displaystyle d(B_2,B_0)\le \delta }$. 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${\displaystyle m(B)\le \delta }$ and${\displaystyle k\ge k_0}$ thenTherefore, by the absolute continuity of${\displaystyle {\lambda }_{k_0}}$ with respect to${\displaystyle m}$, and since${\displaystyle ϵ}$ is arbitrary, we get that${\displaystyle m(B)\to 0}$ implies${\displaystyle {\lambda }_n(B)\to 0}$ uniformly in${\displaystyle n.}$ In particular,${\displaystyle m(B)\to 0}$ implies${\displaystyle \lambda (B)\to 0.}$

By the additivity of the limit it follows that${\displaystyle \lambda }$ isfinitely-additive. Then, since${\displaystyle \underset{m(B)\to 0}{lim}\lambda (B)=0}$ it follows that${\displaystyle \lambda }$ is actually countably additive.

• 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

• Theorems in measure theory

• CS1 German-language sources (de)
• CS1 Italian-language sources (it)