The answer is no.
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 make a simple experiment with a 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 to work with Lebesgue integrable functions. Future editions of Mathematica may tackle with more general concepts of integrability but we should realize that there will always be some limitations of algoritmical approach to integration of real functions.