Subsets of a sigma finite measure are sigma finite measures.

Question

In Royden's text of real analysis, 2nd edd. There is a statement: Any measurable set contained in a set of sigma finite measure is itself of sigma finite measure, and the union of a countable collection of sets of sigma finite measure is again of sigma finite measure. How can I prove it using a countable collection of measurable sets?

Answer 1

Sketches:

• For the first question, call $X$ the space, $B$ the set and consider $\bigcup_{n\in\Bbb N} A_n=X$ and $B_n=B\cap A_n$.

• For the second question, any $E_n$ is the union of a sequence $\{A^k_n\}_{k\in\Bbb N}$ such that $\mu(A^k_n)<\infty$ for all $k$. Consider $B_n=\bigcup\limits_{1\le h\le n\\ 1\le k\le n} A_h^k$ and prove that $\bigcup\limits_{n\in\Bbb N} B_n=\bigcup\limits_{n\in\Bbb N} E_n$.