prove that $\mu$ is a measure $\sigma$-finite.

Question

Let $A \subseteq \mathbb{R}$ a countable subset. Let $a: A \rightarrow \mathbb{R}^{+}$ any function. For $E \subseteq \mathbb{R}$ define $\mu(E)=\sum_{x\in E\cap A} a(x)$.

(a) Prove that $\mu$ is a measure $σ$-finite on $(\mathbb{R},P(\mathbb{R}))$

(B) find Lebesgue decomposition $\mu= \mu_{a}+ \mu_{s}$ of $\mu$ with respect to the Lebesgue measure $\lambda$. ($\mu_{a} \ll \lambda$ and $\mu_{s}\perp \lambda$)

Answer 1

Hints:

a) $A$ is countable and $a(x)$ is finite.

b) $\mu$ is concentrated on $A$ and $\lambda(A) = 0$.