Problem 3

Exercise 6.2 (Borel-Cantelli)   Consider the measure space $(X,\mathcal{M}, \mu)$ with $\mu$ a positive measure. Let $\{E_k\}$ be a countable family of measurable sets satisfying

$$\sum_{k=1}^\infty \mu(E_k) < \infty.$$ (6.15)

Define

$$E := \{x\in\mathbb{R} \;\vert\; x\in E_k k \}.$$ (6.16)

Prove the following:

1. $E$ is measurable
2. $\mu(E)=0$.

Proof. For part (a), we can construct $E$ from $\sigma$-algebra operations:

$$E = \bigcap_{m=1}^\infty \bigcup_{k=m}^\infty E_k$$ (6.17)

For part (b), the definition of a converging series from basic real analysis tells us that

$$\lim_{m\to\infty}\sum_{k=m}^\infty \mu(E_k) = 0.$$ (6.18)

Intersections are decreasing and the converging series guarantees the first set has finite measure from the following estimate

$$\mu\left(\bigcup_{k=1}^\infty E_k\right) \leq \sum_{k=1}^\infty \mu(E_k)<\infty$$ (6.19)

Now apply continuity from above:

$$\mu(E) = \lim_{m\to\infty} \mu\left(\bigcup_{k=m}^\infty E_k\right) \leq \sum_{k=m}^\infty \mu(E_k)$$ (6.20)

where the inequality follows from the countable subadditivity of measure. Applying the limit $m\to\infty$ on either side yields the desired result. $\qedsymbol$

See Exercise for a proof invoking monotonicity of measure.