As the title suggests, I am trying to prove$$ E|X + Y| < \infty \implies E|X| < \infty $$for independent $X$ and $Y$ (but not necessarily equal in distribution). Triangular inequality goes the other way, i.e.$$E|X + Y| \leq E|X| + E|Y|$$in general. Independence must be the key here somehow.
Something I tried is to prove $E|X - Y| < \infty$. If this is true, then we could use$$2|X| = |(X + Y) + (X - Y)| \leq |X + Y| + |X - Y|$$to show that the expectation of $|X|$ is indeed finite. However, I cannot think of a theorem that allows us to conclude the distance is finite if the absolute sum is finite for independent random variables. Any help or insight is appreciated!