Let $\mathcal{A}$ be an open bounded non-empty subset of $\mathbb{R}^{2}$. We denote by $C(\mathcal{A})$ the set of continuous functions $f$ from $\mathbb{R}^{2}$ to $[0,1]$ such that $\forall (x,y) \in \mathbb{R}^{2} \setminus \mathcal{A},\, f(x,y) = 0$ (in other words $f$ is zero outside $\mathcal{A}$). Show that the supremum
$$\sup_{f \in C(\mathcal{A})} \iint_{\mathbb{R}^{2}} f(x,y)\,dx\,dy$$
exists and defines a real number denoted $V(\mathcal{A})$.