I have read in several places an expression like $f\in \mathcal{M}(\mathbb{R}^N)\cap L_p(\mathbb{R}^N)$, where $\mathcal{M}(\mathbb{R}^N)$ denotes the space of Radon measures with finite total variation and $L_p$ is the usual Lebesgue space with $1\leq p \leq \infty$. In general, I refers to expressions of the form $f\in V \cap W$, where $V$ is a measure space and $W$ is a space of functions. My question is: In what sense we say that a function belongs to $\mathcal{M}(\mathbb{R}^N)$?, that is, how we identify a measure with a function?
Maybe, this question sounds evident but I am little lost with this theory.
The basic intuition is that you can think of a function as a density with respect to the Lebesgue measure, $m$. That is, given a measurable positive function $f$, you can define a measure by $$\mu(A) = \int_Afdm.$$ And conversely given a measure $\mu$ that is absolutely continuous with respect to $m$ (which means $m(A) = 0 \implies \mu(A) = 0$) The Radon-Nikodym theorem states that such an $f$ exists.
The proper theoretical framework for unifying functions and measures is Distribution Theory. You may also find this short article by Terence Tao to be of interest.