Let $X$ be a projective, smooth variety over $\mathbb{C}$. Let $Y$ in $X$ be a possibly singular closed subvariety of dimension $k$.
Given $\omega \in H^k(X)$, we can restrict $\omega$ to the smooth locus of $Y$ and integrate. I think (but I am not sure) that this is always finite, and in particular defines a class of $H^k(X)^*$. Is this a correct definition of the fundamental class $[Y]$ of $Y$ (which could also be obtained by triangulating $Y$)?
Why is the integral $\int_{Y \setminus Y_{Sing}} \omega$ always finite? Is it?
A big machinery way to approach this would be to consider the resolution of singularities $\tilde{Y} \to Y \to X$, and pullback $\omega$ to $\tilde{Y}$ instead and integrate, knowing that the difference is essentailly a set of measure zero because $\tilde{Y} \to Y$ is a birational. But this is way overpowered!
(Really I am trying to prove that if $\phi : X \to X'$ is a regular map, then $[\phi^{-1}(Y)] = \phi^*[Y]$, thinking of the fundamental class as linving in cohomology after using universal coefficients. From the definition outlined above I think it is a consequence of Fubini's theorem, but I might be mistaken...)
I would appreciate a good reference!