I know that, if measure $\mu$, with which measure space $X$ is endowed, has a countable base, i.e. if for any measurable $M\subset X$ there exists a measurable set $A_k\in\mathscr{A}$, where $\mathscr{A}$ is a countable family of sets, such that $\mu(M\triangle A_k)<\varepsilon$, then space $L_2(X,\mu)$ of the equivalence classes of Lebesgue square integrable functions is separable. Moreover $L_2(X,\mu)$ is Euclidean and complete and therefore, if it has an infinite dimension, is a separable Hilbert space, isomorphic to the space of square summable sequences, $\ell_2$, either complex or real concondingly with $L_2(X,\mu)$.
I read (p. 386 here) in Kolmogorov-Fomin's that $\ell_2$ can be considered as $L_2(X,\mu)$ for $X$ countable [which I think to mean that $\mu$ has a countable base] and $\mu$ defined on all its subsets and taking value $1$ on every point.
Does anybody know what it means? If I take $X:=\mathbb{R}$ and Lebesgue linear measure as $\mu$, $X$ ha measure $\mu$ with a countable base, $L_2(X,\mu)$ is $\infty$-dimensional, but $\forall t\in\mathbb{R}\quad\mu\{t\}=0$, therefore I have misunderstood the meaning of that sentence. I thank you all for any answer!!!
No, they do mean that $X$ is a countable [countably infinite] set, and the measure is the counting measure on $X$,
$$\mu(A) = \operatorname{card} A$$
for any subset $A\subset X$. In particular, the $\sigma$-algebra of $\mu$-measurable sets is $\mathfrak{P}(X)$, the entire power set of $X$, and the only null set is $\varnothing$.
Thus for $f\colon X \to \mathbb{C}$ and $A\subset X$, we have
$$\int_A f\,d\mu = \sum_{a\in A} f(a)$$
when the sum exists, and up to the identification of a function with its equivalence class modulo "equal almost everywhere", we have
$$L^2(X,\mu) = \left\{ f\colon X \to \mathbb{C} : \int_X \lvert f\rvert^2 < \infty\right\} = \left\{ f\colon X\to\mathbb{C} : \sum_{x\in X} \lvert f(x)\rvert^2 < \infty\right\} = \ell^2(X).$$
Any bijection between two countably infinite sets is measure-preserving for the counting measures on the respective sets, so all these spaces are isomorphic to the classical $\ell^2(\mathbb{N})$.