Inequality on a domain and on compact subsets of the domain

Question

Let $u \in L^1_{loc}(\Omega)$ on a bounded domain $\Omega$. Suppose $$u > 0 \quad \text{a.e. on compact subsets of $\Omega$}.$$ Does this imply that $$u > 0 \quad \text{a.e. on $\Omega$}?$$ This is trivially true, isn't it? Because we can exhaust almost every point of $\Omega$ by compact subsets. The two statements are equivalent?

Answer 1

Yes it does. It's immediate from the inner regularity of Lebesgue measure $\lambda^d$.