Conditional expectation and almost sure equality for integrable random variables

Question

I am given two integrable random variables $X$ and $Y$, where $\mathbb{E}(X\mid Y)=Y$ almost surely (a.s.) and $\mathbb{E}(Y\mid X)=X$ a.s. If the variances of $X$ and $Y$ exist, I can prove easily that $X=Y$ a.s. if I use the Cauchy-Schwarz-inequality. My question is, does this result also hold for integrable random variables? And if so, why?