Given a topological space $X$ we can generate a sigma algebra $\mathcal{M}$ on $X$ by taking the smallest sigma algebra containing all the open sets in $X$.
I wonder if in this situation it is possible to fully characterize a measure $\mu$ on $\mathcal{M}$ by defining its values on open sets in $X$.
I suspect it might be, but I might be wrong so I'd like a confirmation, or maybe to be even more specific on can we characterize a measure by defining $\mu$ on a basis for the topology in $X$ (or subbasis)?
As a special case consider $X = \mathbb{R}$ and as open $\left(a,b\right), -\infty \leq a \leq b \leq + \infty$, can I characterize a measure by simply defining $\mu(\left(a,b\right))$?