Is $x \mapsto \mu( \text{supp}f_x)$ continuous?

Question

Let $X$ be a measurable space equipped with a complex Borel measure $\mu$, and let $\{f_x\}_{x\in X}$ be a family of continuous functions $f_x:X\rightarrow \mathbb{C}$ such that the function $X\rightarrow \mathbb{C} (x\mapsto f_x(y))$ is continuous for all $y\in X$. We set $$\text{supp}f_x=\{y\in X: f_x(y)\neq 0\}.$$ Then is the function $F:X\rightarrow\mathbb{C}(x\mapsto \mu(\text{supp}f_x))$ integrable? If not, what is an assumption for the space $X$ (or the measure $\mu$) for which it holds? Furthermore, is the function $F$ continuous?