Tuesday, July 3, 2018

probability - Clarification of "in" in omegainOmega:X(omega)leqxinmathcalF for a random variable

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:ΩR with the property that
{ωΩ:X(ω)x}F
for each xR. Such a function is said to be F-measurable.




Q1: Because of the curly brackets I guess {ωΩ:X(ω)x} is a set, right?



Q2: I know if a is an element of the set A we write aA. But if a set B is a subset of a set C, we write BC and not BC.



So if {ωΩ:X(ω)x} is a set shouldn't we use "" instead of "", i.e.
{ωΩ:X(ω)x}F?

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