I am reading a proof of the HJB equation (Stochastic Optimal Control) on the book by Oksendal (Chapter 11) and I encountered this:
Given $Y_t=(s+t,X_{s+t})$ and $G$ a fixed domain in $\mathbb{R}\times \mathbb{R}^n$. Additionally, $X_s =x$, so $Y_0=(s,x)=:y$.
Let $W\in G$ be of the form $W=\{(r,z)\in G; r<t_1\}$ for some $t_1>s$. Put $\alpha=\inf \{t\ge 0: Y_t\in W\}$.
Eventually, we have this statement, for a continuous $F$ on $y=(s,x)$:
$\mathbb{E}^y\left[\int_0^{\alpha}F(Y_r)dr\right]\le 0$. And since $\mathbb{E}^y[\alpha]>0$, $\dfrac{\mathbb{E}^y\left[\int_0^{\alpha}F(Y_r)dr\right]}{\mathbb{E}^y[\alpha]}\le 0$.
Then the proof says: letting $t_1 \to s$, we have $F(y)\le 0$ since $F$ is continuous at $y$. Here's where I got lost. How did this happen?
I believe when we let $t_1\to s$, then $\alpha \to 0$. But how come we can conclude that the integrand $F(y)\le 0$?