# Let $X$ and $Y$ be non negative i.i.d random variables such that $E[X]<\infty$. Show that $E[min(X,Y)^2]<\infty$

Let $$X$$ and $$Y$$ be non negative i.i.d random variables such that $$E[X]<\infty$$. Without assuming that $$E[X^2]<\infty$$, show that $$E[\min(X,Y)^2]<\infty$$.

We defined the expectation as: Let $$X:\Omega\rightarrow S$$ be a random element of $$(S,\mathcal{S})$$ with distribution $$\mu$$ and let a measurable function $$h:S\rightarrow \mathbb{R}$$ then $$E[h(x)]=\int_{S}h(x)\mu(dx)$$ whenever LHS or RHS are well defined.

First of all I am actually clueless on how to proceed to solve the question. But this might be due to the fact that I'm not so comfortable with this definition. I don't really understand what $$\mu$$ is and what does $$\mu(dx)$$ represent.

I also feel like we should suppose that $$E[Y]<\infty$$ as well.

$$E[Y] < \infty$$ is implied because $$X$$ and $$Y$$ are i.i.d $$\mu(dx)$$ represents the measure of the infinitesimal segment $$dx$$ according to the probability distribution of $$p$$. Probability distributions cannot always be represented with a function, but if they can you might see this written as $$p(x) dx$$.
Finally, $$\min(x,y)^2 \leq x y$$, thus $$E[\min(x,y)^2] \leq E[x y] = E[x]E[y] < \infty$$.