Let $\sigma_x$ be a measure on $\mathcal{P}(\mathbb{R^n})$.
$\sigma_x(F):=\begin{cases}1,&\text{if }x \in F\text{ }\\0,&\text{otherwise } \end{cases}\quad$
How to prove that all maps $g:\mathbb{R} \to \mathbb{R}$ are integrable with respect to $\sigma_x$?
I know that $\int_\mathbb{R}g=g(x)$.
I tried to use that:
$\int_\mathbb{R}|g| d\sigma_x= |g(x)|<\infty$ so all $g$ are integrable with respect to $\sigma_x$
But I'm not sure if this works.