I have a sequence of functions $f_n(x) \rightarrow f(x) = \frac{g(x)}{x}$ pointwise almost everywhere such that $|f_n(x)| \leq \left|2\frac{g(x)}{x}\right|$ and the integral $\int 2\frac{g(x)}{x} ~dx$ exists as a Cauchy Principal value. (In fact $g(x) \in C^\infty_c(\mathbb{R}^n$))
Now I want to conclude $\int f_n(x) ~dx \rightarrow \int f(x) ~dx = \int \frac{g(x)}{x} ~dx$, but classical Lebesgue dominated convergence doesnt allow me to, because obviously $\int \left|\frac{g(x)}{x}\right| dx$ doesn't exist. Only $\int \frac{g(x)}{x} ~dx$ does, and also only as a Principal Value integral.
Are their variants of the Lebesgue dominated convergence theorem, that allow me to draw the desired conclusion or can I conclude it from Lebesgue with a bit more of work?