X is a random variable so is |X|?

[BoogieE]

Howdy guys. Given that X is a random variable how would you prove |X| to be one too? Thanks for any suggestions!

 

[micromass]

What is your definition of random variable?

 

[BoogieE]

Just mapping from (S,F) to (R, B(R))

 

[micromass]

X is measurable, | | is continuous hence measurable. And the composition of measurables is measurable.

 

[BoogieE]

I am expecting that from the fact that X(-1)(G) = {w in S such that X(w) is in G for all G in B(R) } is in F you can somehow show that |X|(-1)(G') = {w in S such that |X|(w) is in G' for all G' in B(R)} is also if F

 

[BoogieE]

X is measurable, | | is continuous hence measurable. And the composition of measurables is measurable.

I asked my math professor and she said this is ok. I probably overthought the problem. Thank you very much!