Exercise 1.4.25 (Measure on a discrete $\sigma$-algebra)

Question

Let $X$ be an at most countable set with a discrete $\sigma$-algebra $\mathcal{B}$. Show that every measure $\mu$ on this measurable space can be uniquely represented in the form
$$\mu = \sum_{x\in X} c_x\delta_x $$
for non-negative real numbers $(c_x)_{x\in X}$. In other words, for any $E \in \mathcal{B}: \mu(E) = \sum_{x\in E} c_x$.

Elu Thingol · Accepted Answer

Denote for all $x\in X$: $c_x :=\mu(\{x\})$. We argue that this $(c_x)_{x \in X}$ are exactly the non-negative real numbers given in the exercise. Now pick an arbitrary $E \in \mathcal{B}$. Any discrete $\sigma$-algebra is an atomic algebra, in our case the atoms are the elements of $X$ themselves. Thus, we can trivially write $E$ as a disjoint union $E = \bigcup_{x \in E} \{x\}$. By the properties of the measure
$$\mu(E) = \mu\left(\bigcup_{x \in E} \{x\}\right) = \sum_{x\in E} \mu(\{x\}) = \sum_{x\in E} c_x = \sum_{x\in E} c_x \cdot 1_E(x) + \sum_{x\in X/E} c_x\cdot 1_E(x) = \sum_{x\in X} c_x \cdot 1_E(x) = \sum_{x\in X} c_x \cdot \delta_x(E).$$

Exercise 1.4.25 (Measure on a discrete $\sigma$-algebra)

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Exercise 1.4.25 (Measure on a discrete $\sigma$-algebra)

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