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?



  • 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 $\bigcup\limits_{n\in\Bbb N} B_n=\bigcup\limits_{n\in\Bbb N} E_n$.

  • $\begingroup$ Can you provide me full part of the proof? $\endgroup$ – temesgen tilahun Sep 3 '18 at 18:37

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.