$L^2$ equality implies $L^1$ equality or a.e. equality?

Question

Suppose $f\in L^1 \cap L^2$ and $g\in L^2$.

If I know that $f=g$ in $L^2$, then does it follow that $f=g$ in $L^1$ or a.e?

Does it follow that $g\in L^1$?

More generally, if $f\in L^s \cap L^r, g\in L^s$ and $f=g$ in $L^s$, then does it follow that $g\in L^r$?

In particular, I have the Lebesgue measure on $\mathbb R$ in mind but I am also curious about general results.

Thanks and regards.