If $f(x,y)$ is integrable and $\int f(x,y) dx \leq 0$ for a.e. $y$, then do we have $f(x,y) \leq 0$ for a.e. $(x,y)$?

Question

Let $f : [0,1]^2 \to \mathbb{R}$ be an integrable function such that \begin{equation} \int_0^1 f(x,y) dx \leq 0 \end{equation} for a.e. $y \in [0,1]$.

Then, I wonder if $f(x,y) \leq 0$ for a.e. $(x,y) \in [0,1]^2$.

I tried to approach by decomposing $f$ into its positive and negative parts, but it seems quite confusing.. Could anyone please help me?

Answer 1

Not at all, let for example $f(x,y)= 1-3x$, then the integrals are negative for all $y\in[0,1]$. But $f>0$ on a set with positive measure.