Relationship involving Lebesgue integral of supremum of function depending on a parameter

Question

Suppose $f:\mathbb{R}^2\to\mathbb{C}$ is continuous and nonnegative. For each $x\in\mathbb{R}$ define $f_x:\mathbb{R}\to\mathbb{C}$ by $$f_x(y)=f(x,y)\qquad(y\in\mathbb{R}).$$ Is there a Fatou's-Lemma-like relationship between $$\int_{-\infty}^\infty\sup_{x\in\mathbb{R}}f_x(y)\,dy$$ and $$\sup_{x\in\mathbb{R}}\int_{-\infty}^\infty f_x(y)\,dy?$$ If so, what would it be, exactly, and how would it be proven. If some other conditions are needed for such a relationship, please feel free to suggest them. What's not alterable is that $x$ can take on uncountably many values.

Answer 1

For any $x$ we have $f_x(y) \le \sup_{x' \in \mathbb{R}} f_{x'}(y)$ for all $y$, so $$\int f_x(y) \, dy \le \int \sup_{x' \in \mathbb{R}} f_{x'}(y) \, dy$$ holds for any $x$. Then take the supremum with respect to $x$ on the left-hand side.