Wrong formulation of Monotone Convergence

Question

In my Probability Theory Lecture we stated the Theorem of Monotone Convergence:

Let $Y, Y_1, Y_2, \dots$ be discrete, integrable, real-valued random variables with $Y_k \uparrow Y$ almost surely , e.g. $Y_1 \leq Y_2 \leq \dots$ a.s and $\lim_{k \rightarrow \infty } Y_k = Y$ a.s. Then we have:

$$\lim_{k \rightarrow \infty} \mathbb{E}[Y_k] = \mathbb{E}[Y]$$

And I'm wondering if this formulation is correct, since I can't find the non-negativity of the $Y_i's$. Am I missing something?

Answer 1

It is correct. Note that $Y_k-Y_1\uparrow Y-Y_1$ a.s. and $Y_k-Y_1\geq 0$ a.s. Applying monotone convergence we get $$\lim_{k\to\infty}\mathbb E[Y_k-Y_1]=\mathbb E[Y-Y_1]. $$Finally use the integrabilities of $Y_k$s and $Y$ to obtain $$\lim_{k\to\infty}\mathbb E[Y_k]=\mathbb E[Y].$$