Is $\mathcal{L}_{M}(\Omega, \mathcal{F}, \mathbb{P})$ a linear subspace?

Question

Is the space $\mathcal{L}_{M}(\Omega, \mathcal{F}, \mathbb{P})$ of of all random variables, $X$, whose moment-generating function $\mathbb{E}[e^{tX}]$ exists for all $t \in \mathbb{R}$ a linear subspace of the set of all random variables ?

I dont think it is because if $X,Y \in \mathcal{L}_{M}(\Omega, \mathcal{F}, \mathbb{P})$ then it does not need to be the case that $X+Y \in \mathcal{L}_{M}(\Omega, \mathcal{F}, \mathbb{P})$. But I cant think of any counterexamples.

Answer 1

$\newcommand{\E}{\mathbb E}$Why not? One has using Cauchy-Schwarz: \begin{align} 0\leq \E[e^{t(X+Y)}]=\E[e^{tX}e^{tY}]\leq \sqrt[]{\E[e^{2tX}]\E[e^{2tY}]}=\sqrt[]{M_X(2t)M_Y(2t)}<\infty \end{align} So $X+Y\in\mathcal{L}_M(\Omega,\mathcal F,\mathbb P)$. We also have $\alpha X\in\mathcal{L}_M(\Omega,\mathcal F,\mathbb P)$ that is even easier to prove. So $\mathcal{L}_M(\Omega,\mathcal F,\mathbb P)$ is a linear subspace.