Upper bound for some measurable sets given the inequality $\sum_{n=1}^{\infty} \mu (A_{n}) \leq \mu (\bigcup_{n=1}^{\infty} A_{n}) + \epsilon$

First note that $B_n$'s are disjoint and their union is $A$. Hence $\sum \mu(A_n) <\sum \mu(B_n)+\epsilon$ which implies $\sum \mu(A_n\setminus B_n) <\epsilon$. Now $\mu(A\cap E)+\epsilon =\sum \mu(B_n \cap E) +\epsilon > \sum \mu(A_n \cap E) $ (because $\sum \mu(E \cap [A_n\setminus B_n]) <\epsilon$). This proves the first part.

For the second part take $E=C_k$ in the first part. Note that $\int \sum_n I_{A_n} I_{C_k} \geq k \int I_{C _k}=k \mu(C_k)$. Hence $k \mu(C_k) \leq \sum_n \mu(A_n\cap C_k) \leq \sum_n \mu(B_n\cap C_k)+\epsilon = \mu (A\cap C_k) +\epsilon \leq \mu(C_k)+\epsilon$. This give part 2).

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Real Analysis

Measure Theory

Measurable Sets

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