Let $\delta_x$ be a measure on $\mathcal{P}(\mathbb{R^n})$.
Define $\delta_x(E)=\begin{cases}1,&\text{if }x \in E\text{ }\\0,&\text{if }x\not\in E\end{cases}\quad$
How can it be shown that every map $f: \mathbb{R} \to \mathbb{R}$ is integrable with respect to $\delta_x$?
I tried to use:
If $f: [a,b] \to \mathbb{R}$ is a Riemann integral, then $f \in \mathcal{L}(E,\delta_x)$.
So: $\int f \ \delta_x= \int_{a}^{b}f$
Here I don't know how to continue.
Or is there another way to prove that all $f: \mathbb{R} \to \mathbb{R}$ are integrable with respect to $\delta_x$?