The answer is no because of fundamental mathematical limitations which origin in the set theory regarding countability (see also Cantor's theorem) - functions over a given set are more numerous than its (power) cardinality. Mathematica cannot integrate even much more restricted class, namely Riemann integrable functions, all Riemann integrals are equal to Lesbegue integrals. The class of functions which could be integrated over a domain in $\mathbb{R}^n$ in Mathematica is of "measure zero" in the class of Lesbegue integrable functions. More precisely we need rather Baire categories to work with general topological concepts of class of adequate functions.
Another thing is that you are going to think of NIntegrate rather than of Integrate.
Let's try to integrate a simple Lebesgue integrable function defined in $\mathbb{R}$.
f[x_] /; x ∈ Rationals && 0 <= x <= 1 := 1
f[x_] /; ! (x ∈ Rationals) && 0 <= x <= 1 := 0
f[Sqrt[3]/2]
f[1/2]
0
1
but neither Integrate nor NIntegrate can calculate adequate integrals:
Integrate[ f[x], {x, 0, 1}]
NIntegrate[ f[x], {x, 0, 1}]
although we know it should be 0.
On the other hand we could always remedy various problems algorithmically but in general it can't be done in full generality (e.g. because of finite number of states of computers) to work with Lebesgue integrable functions. Future editions of Mathematica may involve more powerful capabilities for tackling integrability problems but we should realize that there will always be some limitations of algoritmical approach to integration of real functions.