# Functions are integrable with respect to a measure

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 $$\delta_x$$?

• "Since $\delta_x$ seems to be a step function", no, it is not; it is a "measure".
– user587192
Nov 21, 2018 at 18:57
• Do you know the definition of $f$ being integrable with respect to $\delta_x$? That should be the first step of doing this problem.
– user587192
Nov 21, 2018 at 18:58

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.