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Vitali–Hahn–Saks theorem

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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

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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

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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

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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

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  • 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