Vitali–Hahn–Saks theorem

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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