[Math] Finite measure space & sigma-finite measure space

Question

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?

Answer 1

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.

edit to add: and I think pizza has said it much better than me.