Functions are integrable with respect to a measure

Question

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 $\delta_x$?

Answer 1

One can directly check the definition of Lebesgue integrability. For any simple function $f$, one can directly verify that the integral is equal to $f(x)$. Then it extends to non-negative measurable $f$ by the definition of Lebesgue integral. If $f$ takes real values, then $\int|f| d\delta_x= |f(x)|<\infty$, so integrability holds.