If $m$ is a Lebesgue measure on $\mathbb{R}^n$, then the wikipedia says
(1) $f$ is locally integrable if $\int_K f dm < \infty$ for every compact set $K$.
But why is this condition equivalent to
(2) $f$ is locally integrable if $\int_B f dm < \infty$ for every bounded measurable set $B$? as defined in Folland's book Real Analysis?
The direction $(2) \rightarrow (1)$ is clear, but what about the other direction?
Both definitions need a correction. $f$ is locally integrable if $\int_K |f| <\infty$ for every compact set $K$ and this is equivalent to the condition $\int_K |f| <\infty$ for every bounded measurable set $K$. You cannot drop the absolute value sign in the definitions. The equivalence is obvious because any compact set is a bounded measurable set and any bounded measurable set is contained in a closed ball which is a compact set.