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.
is a measure space with
of complex measures.
is absolutely continuous with respect to
the finite limits exist
Then the absolute continuity of the
lim
lim
is countably additive on
Given a measure space
a distance can be constructed on
the set of measurable sets
This is done by defining This gives rise to a metric space
by identifying two sets
with the metric defined above is a complete metric space.
χ
χ
This means that the metric space
can be identified with a subset of the Banach space
Then we can choose a sub-sequence
lim inf
the limit inferior of the sequence) and hence
defines a function
This function is well defined, this is it is independent on the representative
due to the absolute continuity of
By Baire category theorem at least one
must contain a non-empty open set of
Therefore, by the absolute continuity of
By the additivity of the limit it follows that
is actually countably additive.