Let $X$ be a real valued random varaible on a probability space ($\Omega , \mathscr F ,\mathsf P)$ Carefully stating any result you use explain why $\exp(c|X|)$ is a real valued random variable for $c>0.$
Assume $\mathsf E[\exp(c|X|)]< \infty $ for some $c>0$ by the same reasons as above, $\exp(tX)$ is a real valued random variable for every real number $t$, show $\exp(tX)$ is integrable on $(\Omega , \mathscr F , \mathsf P)$ for all $t\in (-c,c).$
For the first part, the bit I am struggling with is the modulus, I know that a continuous function on a probability space is a random variable but I though |X| was not continuous?
For the second part, is it correct to say that |exp(c|X|)| $\geq$ |exp(tX)| for t$\in $(-c,c) so therefore the integral |exp(c|X|)| $\geq$ Integral |exp(tX)| and since the integral of the LHS is finite because we know exp(c|X|) is intregable this means that the integral of the RHS is finite too, thus exp(tX) is integrable ?
This is not a homework question, it is a question from a mock paper to which there are no solutions.