If $\int f=0$ then $f=0$ a.e. with $f\geq 0$. Is it true if $f(x) = \infty$?

Question

I have a doubt!

I know that if $f$ measurable and nonnegative, $\int f=0$ implies $f=0$ a.e.

And if $m(E)=0$ then $\int_{E}f=0$ (even if $f(x)=\infty$ forall $x$)

If $f(x)=\infty$ forall $x$, $f:X\to \overline{\mathbb{R}}$ and $m(X)=0$ then $\int_{X}f=0$ implies $f=0$ a.e.

Can it be $f = 0$ a.e. even when $f(x) = \infty$ for all $x$ in $X$?