Measures of translates of an open set

Solution 1:

For the sake of having an answer, I'll note that Mike's comment about Fatou's lemma works.

Take a sequence $x_n \to x$. Note that $\liminf 1_{V + x_n}(y) \ge 1_{V + x}(y)$ for every $y$. In other words, if $y \in V + x$, then $y \in V + x_n$ for all but finitely many $n$. The converse need not hold, so the inequality can be strict for some $y$, but that is okay.

Now Fatou's lemma says $$\liminf \int 1_{V + x_n} \ge \int \liminf\, 1_{V + x_n} \ge \int 1_{V + x}$$ which is to say $\liminf \varphi(x_n) \ge \varphi(x)$.

This does not require any assumptions on the measure, except that it be positive.