# Finite measure space & sigma-finite measure space

A measure space $$(X, \Sigma, \mu)$$ is finite if $$\mu(X)<\infty$$.

It is equivalent to saying that $$(X, \Sigma, \mu)$$ is finite if $$\mu(E)<\infty$$ for all $$E \in \Sigma$$

A measure space $$(X, \Sigma, \mu)$$ is $$\sigma$$-finite if X is a countable union of sets with finite measure.

1. Does $$\sigma$$-finiteness imply that $$\mu(E)<\infty$$ for all $$E \in \Sigma$$?
2. If $$\mu(E)<\infty$$ for all $$E \in \Sigma$$, dose it imply $$\sigma$$-finiteness or finiteness of a measure space?
• Do you know any examples of $\sigma$-finite measure spaces?
– user147263
Apr 17 '15 at 4:03
• I know that Real numbers with Lebesgue measure is sigma finite, but not finite just by taking all unit intervals [k, k+1] etc Apr 17 '15 at 4:08
• Not finite because $\mu(\mathbb{R})=\infty$. So that's your answer to 1.
– user147263
Apr 17 '15 at 4:08
• I think both of your questions would be resolved at once if you recalled that $X$ itself is an element of $\Sigma$.
– user147263
Apr 17 '15 at 4:09
• oh yea, I see it. Thanks pizza! Apr 17 '15 at 4:11

Probably the best example of a finite measure space is $[0, 1]$ with its usual structure, and the best example of a $\sigma$-finite measure space is $\mathbb{R}$ with its usual structure. So, are all the measurable subsets of $\mathbb{R}$ finite in measure? That should answer your first.
For your second, consider what $\mu(X) < \infty$ implies.