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
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|
(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.
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