There're many versions of Riesz representation theorems saying that certain functionals $T$ on certain spaces $X$ can be represented by integrals of the form
$$ Tf = \int_X fd\mu $$
Where $\mu$ is some measure with certain features. For example Haar measures for locally compact groups, regular measures when dealing with $C(X)$, or $fd\mu$ with $f \in L^q(X,\mu)$ when the vector space is $L^p(X,\mu)$ (with $q$ conjugate exponent to $p$).
A question I have, for which I cannot find an answer is whether or not it is possible to characterize such measures by value on some "basis".
In finite dimensions a linear transformation is fully defined by the images over a basis for example. Basis for vector spaces of infinite dimensions are another story... I don't think it's that straightforward.
I am interested to know if $\sigma$-algebras have some notion of basis, just like topological spaces, and maybe there's some uniquiness theorem characterizing a measure over such basis.
I do know if we have a semi-algebra we can define the algebra generated by such sigma algebra and given an algebra we can define the sigma algebra generated by such algebra, I don't know if this is sufficient to characterize completely a both a sigma algebra and the measures over such algebra.
