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
(Almost sure convergence 2)
The key difference from the almost sure convergence is the
We use the
for uniform convergence. Using the technique of the section
Operations on sets and
) we state that the set
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
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.