Let $(X_n)_{n\geq 1}$ be a sequence of non negative, i.i.d random variables
and $S_0:=0$, $S_n:=\sum_{k=1}^n X_k$. Further let $(\Omega,F,\mathbb{P})$ be a probability space so large it contains all the information of the sequence $(X_n)$ and let $\tau$ be a $F_k:=(\sigma \{ S_0,...,S_k \} )_k $ stopping time, with $\mathbb{E} \tau < \inf$, Then $\mathbb{E}S_\tau=\mathbb{E}\tau \mathbb{E} X_1$.
Now this is a special case of Wald's idendity. Im wondering about one point in the proof, the author argues like that: $\mathbb{E}S_\tau=\mathbb{E} [\sum_{k \geq 1} X_k \mathbb{1}_{ \{ \tau \geq k \} } ]$ = $\sum_{k \geq 1} \mathbb{E} [ X_k \mathbb{1}_{ \{ \tau \geq k \} } $]
He argues, that the last equality is an application of Fubini which is possible because $ X_k \mathbb{1}_{ \{ \tau \geq k \} }$ is non-negative. I dont see how Fubini is used here, since we only have 1 integral. Isnt it possible to just use monotone convergence theorem?