# Calculate Delta measure

We are looking at the measure space $$(\mathbb{R}^d, \mathcal{P}(\mathbb{R}^d),\delta_p)$$, for all $$A\subset \mathbb{R}^d$$:

$$\delta_p(A):=\begin{cases} 1, & \text{if } p \in A \\ 0, & \text{otherwise} \end{cases}$$.

1. Let $$A \subset \mathbb{R}^d$$ and $$f:\mathbb{R}^d \rightarrow \overline {\mathbb {R}}$$ arbitrary. Show that $$f$$ is measurable and calculate $$\int_A f \space d \delta_p$$.

2. Let $$\lambda$$ be the Lebesgue measure on $$\mathbb{R}^d$$. Does an integratable function $$h: \mathbb{R}^d \rightarrow \mathbb{R}$$ exist such that for $$A \subset \mathbb{R}^d: \delta_p(A)=\int_Ah \space d \lambda$$?

$$\int_{A}fd\delta_p= \begin{cases} f(p) & \text{if}~ p\in A\\ 0 & \text{otherwise} \end{cases}$$ is the integral but how do I show this rigorously?

• For 1. note that for any set $A$ we must have $f^{-1} (A) \in \mathcal{P}(\mathbb{R}^d)$. Nov 30, 2020 at 17:31

Use the fact that, for an increasing sequence $$(f_n)_n$$ of simple functions with $$\lim_{n \to +\infty}f_n(x) = f(x)$$, $$\int f~d\delta = \lim_{n \to \infty} \int f_n~d\delta$$ by definition of the integral. (Note that in the definition of the integral we suppose $$f \ge 0$$, you just have to separate $$f^+$$ and $$f^-$$).