I saw an amazing (I think) tool in one of the exercises of Durrett (page 225) which I am going to use it as a lemma to solve another (amazing) problem. The Durrett's is:
Ex4.1.4. Suppose $X\geq 0$ and $EX= \infty$. Show that there is a unique $F$-measurable $Y$ with $0\leq Y\leq\infty$ so that
$\int_A X \,dP = \int_A Y \,dP $ for all $A\in F$.
With its hint this problem becomes easy. But now my problem is if we have $Z_1, Z_2, Z_3$ and $Z_4$, four random variables where none of them have bounded expectation, and we know that $E[Z_1 + Z_2] = 0$ and $E[Z_3 + Z_4] < \infty$ and that $Z_1 \stackrel{d}{=} Z_3$ and $Z_2 \stackrel{d}{=} Z_4$. Can we now conclude that $E[Z_3 + Z_4]= 0$?