I have a basic question regarding the definition of a random variable. Probability and Random Processes (Grimmett and Stirzaker) have the following:
A random variable is a function X:Ω→R with the property that
{ω∈Ω:X(ω)≤x}∈F
for each x∈R. Such a function is said to be F-measurable.
Q1: Because of the curly brackets I guess {ω∈Ω:X(ω)≤x} is a set, right?
Q2: I know if a is an element of the set A we write a∈A. But if a set B is a subset of a set C, we write B⊂C and not B∈C.
So if {ω∈Ω:X(ω)≤x} is a set shouldn't we use "⊂" instead of "∈", i.e.
{ω∈Ω:X(ω)≤x}⊂F?
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