Hie everyone, I need a check on the following passage I found in a proof:
Let $X$ be an a.s. positive random variable and let $h,t>0$. Then $$\lim_{h \rightarrow 0^{+}} E[\frac{e^{-hX} - 1}{h}] = E[\lim_h \frac{e^{-hX} - 1}{h}]$$
Of course I need to apply a theorem such as Dominated Conv. theorem or Monotone convergence theorem, and I need a check on my reasoning.
If I look at the graph of the function (of $h$ ) $$f_X(h) = \frac{1-e^{-hX}}{h}$$ I can see that it converges monotonically to $0$ but the sequence is decreasing... so I can't apply the monotone convegence theorem in order to take the limit inside the expectation.
BUT If I look at the graph again, I can see that it's bounded by $X$ but $X$ is random and no one told me it's integrable. So I don't think I can apply dominated convergence.
I think I should apply monotone convergence, but the seuqence is not incresing... what can I do?