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
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
Then
This means that the metric space
can be identified with a subset of the Banach space
.
Let , with
Then we can choose a sub-sequence
such that
exists almost everywhere and
. It follows that
for some
(furthermore
if and only if for
large enough, then we have that
the limit inferior of the sequence) and hence
Therefore,
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
is closed in
, and by the hypothesis
we have that
By Baire category theorem at least one
must contain a non-empty open set of
. This means that there is
and a
such that
implies
On the other hand, any
with
can be represented as
with
and
. This can be done, for example by taking
and
. Thus, if
and
then
Therefore, by the absolute continuity of
with respect to
, and since
is arbitrary, we get that
implies
uniformly in
In particular,
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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