Distributions have a remarkably simple definition: given a paracompact smooth manifold $M$, a distribution on $M$ is a linear transformation $\mathcal{C}^\infty_{\mathrm{cs}}(M) \rightarrow \mathbb{R},$ where $\mathcal{C}^\infty_{\mathrm{cs}}(M)$ is the set of compactly-supported smooth functions $M \rightarrow \mathbb{R}$.
Question. Can measures on a measureable space be defined in a similar way?
Something like this: let $(X,\mathcal{A})$ denote a measureable space. Then a measure on $(X,\mathcal{A})$ is an $[0,\infty]$-linear function $F(X,\mathcal{A}) \rightarrow [0,\infty]$ such that [whatever], where $F(X,\mathcal{A})$ for the set of $\mathcal{A}$-measurable functions $X \rightarrow [0,\infty]$.
I was thinking that maybe the monotone convergence theorem is the correct [whatever]. Thoughts, anyone?