he concept of almost sure convergence was defined by the formula
(
Almost sure convergence
).
Equivalently, we characterize the set
where the random variables
converges to the random variable
by the
relationship


(Almost sure convergence 2)

The key difference from the almost sure convergence is the
independence from
.
We use the
notations
for uniform convergence. Using the technique of the section
(
Operations on sets and
logical statements
) we state that the set
satisfies


(Uniform convergence)

for some function
.
Uniform convergence implies the convergence a.s. The inverse relationship is
given by the following Egorov's theorem.
To prove the Egorov's theorem we need the continuity property of the
additive
measure. It serves as a bridge between the convergence concepts defined in
terms of set algebra and the topology introduced by the probability measure.
We now proceed with the prove of the Egorov's theorem.
