Is it always possible to define a non-degenerate, finite measure on a given $\sigma$-algebra?

Question

1. Does every $\sigma$-algebra admit some non-degenerate (i.e. not identically zero), finite measure?
2. Does every $\sigma$-algebra admit some non-degenerate, finite content (i.e. a finitely-additive, but not necessarily $\sigma$-additive measure)?

Answer 1

Yes. If by $\sigma$-algebra you mean a sigma algebra of subsets of a set $X$ you automatically get that for all points $x\in X$ all Dirac delta $\delta_x$ are measures. This answers even the second question.