Are continuous functions of random variables, random variables too?

For example, $\displaystyle e^E$, for $\displaystyle E$ being a random variable. Is it a r.v?

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- Apr 30th 2010, 10:42 AMgalahadFunctions of Random Variables
Are continuous functions of random variables, random variables too?

For example, $\displaystyle e^E$, for $\displaystyle E$ being a random variable. Is it a r.v? - Apr 30th 2010, 01:39 PMGnomeSain
- May 1st 2010, 04:00 AMMoo
Hello,

A random variable is a measurable application (from a space to another).

A continuous function is measurable *provided you are in a correct sigma-algebra, as Laurent pointed out below ^^.

The composition of two measurable functions is measurable.

Thus the composition of a continuous function and a random variable is a random variable (Tongueout) - May 1st 2010, 04:09 AMLaurent
To be rigorous, you should specify which $\displaystyle \sigma$-algebras you endow the probability spaces with. What always holds true is that measurable functions of random variables are random variables. If, as usual, the spaces are endowed with the Borel $\displaystyle \sigma$-algebra (or its completion), then continuous functions are indeed measurable so your statement is correct.