Existence of function nonnegative Lebesgue integrable such that $|f(x,y)| \leq g(x)$

Question

Let $f:[0,1] \times \mathbb{R} \to \mathbb{R}$ continous function is it true that there is $g$ Lebesgue integrable nonnegative such that $|f(x,y)| \leq g(x)$ for all $(x,y) \in [0,1] \times \mathbb{R}$?

I need this fact to solve a problem, using the theorem of the dominated convergence, but i can not show existence of such $g$.

Answer 1

Not generally. If $f(x,y)=e^y,$ then you cannot bound that by a function independent of $y$, as it’s not bounded in that variable. Since $y$ ranges over $\mathbb{R}$, it seems to me like you’d need $f$ bounded in that variable.