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
is a measure space with
and a sequence
of complex measures. Assuming that each
is absolutely continuous with respect to
and that a for all
the finite limits exist
Then the absolute continuity of the
with respect to
is uniform in
that is,
implies that
uniformly in
Also
is countably additive on
Preliminaries
Given a measure space
a distance can be constructed on
the set of measurable sets
with
This is done by defining
where
is the symmetric difference of the sets ![{\displaystyle B_{1},B_{2}\in {\mathcal {B}}_{0}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcbdbe8d5ab6f458d04b2d23a77876889f5bdce8)
This gives rise to a metric space
by identifying two sets
when
Thus a point
with representative
is the set of all
such that
Proposition:
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f233ef0db2638a82921d505535b25219b8e05cbd)
Then
![{\displaystyle d(B_{1},B_{2})=\int _{S}|\chi _{B_{1}}(s)-\chi _{B_{2}}(x)|dm}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e91bbb124256b751843c3446b030070a3d6ae8)
This means that the metric space
![{\displaystyle {\tilde {{\mathcal {B}}_{0}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45deef4655b4301773569e735391de9e84608432)
can be identified with a subset of the Banach space
![{\displaystyle L^{1}(S,{\mathcal {B}},m)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dee42fb945dd41e0ed9b7a77a959e5eaee268873)
.
Let
, 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/603200bd38cd5defd41627ec10aa6fa65c044cd8)
Then we can choose a sub-sequence
![{\displaystyle \chi _{B_{n'}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cacef0fe6612072b929ffb36bf35b480fc4a9144)
such that
![{\displaystyle \lim _{n'\to \infty }\chi _{B_{n'}}(x)=\chi (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47e7cf73bc3e6b1c9efa1e985cf943cbbbacf125)
exists almost everywhere and
![{\displaystyle \lim _{n'\to \infty }\int _{S}|\chi (x)-\chi _{B_{n'}(x)}|dm=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85135fabfc06672f7ed5457eafbcb9217a8f5853)
. It follows that
![{\displaystyle \chi =\chi _{B_{\infty }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa6b62c406838e03d970b2089c97572e25cf27b2)
for some
![{\displaystyle B_{\infty }\in {\mathcal {B}}_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/456601eccb8120e1e96680877893a0bf59e4e3ca)
(furthermore
if and only if ![{\displaystyle \chi _{B_{n'}}(x)=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7de0f771606c3fa6acd0eebb90cb60431795059)
for
![{\displaystyle n'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d215ec5b3d3b48ac8ec46e7131e7b3c091c9114e)
large enough, then we have that
![{\displaystyle B_{\infty }=\liminf _{n'\to \infty }B_{n'}={\bigcup _{n'=1}^{\infty }}\left({\bigcap _{m=n'}^{\infty }}B_{m}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0fb1d4540d235f91d4748915742013ecae2f86a)
the limit inferior of the sequence) and hence
![{\displaystyle \lim _{n\to \infty }d(B_{\infty },B_{n})=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/242381650d8093220612805497fe0917d9a52666)
Therefore,
![{\displaystyle {\tilde {{\mathcal {B}}_{0}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45deef4655b4301773569e735391de9e84608432)
is complete.
Proof of Vitali-Hahn-Saks theorem
Each
defines a function
on
by taking
. This function is well defined, this is it is independent on the representative
of the class
due to the absolute continuity of
with respect to
. Moreover
is continuous.
For every
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 \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a4a1439267023be523058c1d24ebbae9bba17df)
is closed in
![{\displaystyle {\tilde {\mathcal {B}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48080fb5f6584d2f066b473dcc1509221a49d150)
, and by the hypothesis
![{\displaystyle \lim _{n\to \infty }\lambda _{n}(B)=\lambda (B)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3db1b83d208bce9214aa67ebdbed5eaad29efca3)
we have that
![{\displaystyle {\tilde {\mathcal {B}}}=\bigcup _{k=1}^{\infty }F_{k,\epsilon }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eede4b666bc4e13ae803ebc48b6508f62dce4289)
By Baire category theorem at least one
![{\displaystyle F_{k_{0},\epsilon }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be29d185b3a50328cdfbdc2743ef586e239cd39e)
must contain a non-empty open set of
![{\displaystyle {\tilde {\mathcal {B}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48080fb5f6584d2f066b473dcc1509221a49d150)
. This means that there is
![{\displaystyle {\overline {B_{0}}}\in {\tilde {\mathcal {B}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15d81bd91f61400c9a25a12ab542bc3d68500be1)
and a
![{\displaystyle \delta >0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/595d5cea06fdcaf2642caf549eda2cfc537958a9)
such that
![{\displaystyle d(B,B_{0})<\delta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/20bf83e8fc6dfa4163cf8f6e82ba880db94fdcb1)
implies
![{\displaystyle \sup _{n\geq 1}|{\overline {\lambda }}_{k_{0}}({\overline {B}})-{\overline {\lambda }}_{k_{0}+n}({\overline {B}})|\leq \epsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fdf43be1662367652f2ab5d217ef2ead283b8a9)
On the other hand, any
![{\displaystyle B\in {\mathcal {B}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0df7ead8ec7e88c3e04464929ae1213bbc1cd13)
with
![{\displaystyle m(B)\leq \delta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/db69598a54989d86759bd7c4d3af97944c29e7c0)
can be represented as
![{\displaystyle B=B_{1}\setminus B_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/036421b29b1d2acd16055ca69f45219053f4968c)
with
![{\displaystyle d(B_{1},B_{0})\leq \delta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ebbcff562bdf582c1fbc468f95d7b6dc7219cbd)
and
![{\displaystyle d(B_{2},B_{0})\leq \delta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/039d58807d6437b985a93aa5222e704d8ee923a2)
. This can be done, for example by taking
![{\displaystyle B_{1}=B\cup B_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df4ffb00639d7213e7d75cb2e01ea4dbb0a1634f)
and
![{\displaystyle B_{2}=B_{0}\setminus (B\cap B_{0})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5db5ad0eef59327513160c9595023a33bf1e3e6)
. Thus, if
![{\displaystyle m(B)\leq \delta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/db69598a54989d86759bd7c4d3af97944c29e7c0)
and
![{\displaystyle k\geq k_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a60745159f780220691a7e0306745179e41491c)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51515c04509bd249aef954ea8b1c08d6a927e427)
Therefore, by the absolute continuity of
![{\displaystyle \lambda _{k_{0}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08266af9602e59a50f205b2d506f1c9e93c6c4d8)
with respect to
![{\displaystyle m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc)
, and since
![{\displaystyle \epsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2)
is arbitrary, we get that
![{\displaystyle m(B)\to 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/363ebfa995a6b242dab912a122470869c2ce5213)
implies
![{\displaystyle \lambda _{n}(B)\to 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c40d543c08a95be1ad659cd90a2019b8979de677)
uniformly in
![{\displaystyle n.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e59df02a9f67a5da3c220f1244c99a46cc4eb1c6)
In particular,
![{\displaystyle m(B)\to 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/363ebfa995a6b242dab912a122470869c2ce5213)
implies
By the additivity of the limit it follows that
is finitely-additive. Then, since
it follows that
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
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