Let $I:L_1([0,1]^2) \rightarrow \mathbb{R}_+$ be a function defined by
$$I(W) = \int_{[0,1]^2} I_0(W(x,y)) dx dy $$
where $I_0(u) = -u\log(u)-(1-u)\log\left(1-u\right)$.
I sketched a proof of the continuity:
Let $W_m \rightarrow W$ be a pointwise convergent sequence.
Hence by dominated convergence theorem (DCT) $||W_n -W||_1 \rightarrow 0$. Since $I_0(\cdot)$ is continuous, then $I_0(W_n(x,y)) \rightarrow I_0(W(x,y))$. And by DCT, we obtain $I(W_n(x,y)) \rightarrow I(W(x,y))$.
Thus, $I(.)$ is pointwisely continuous. Hence $I(.)$ is continuous on $L_{1}\left(\left[0,1\right]^2\right)$.
Is it correct this proof?