In mathematics, the Vitali–Hahn–Saks theorem, introduced by , , and , 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 Theorems in measure theory