I shall state the dominated convergence theorem:
$\textbf{Dominated convergence theorem: }$Let $f_k:\mathbb{R}^n\to \mathbb{C}$ be a sequence of integrable functions. Suppose there is a positively integrable function $g:\mathbb{R}^n\to[0,\infty]$ such that for all $x\in\mathbb{R}^n$ and all $k$ we have $|f_k(x)|\leq g(x)$.
If $f_k\to f$ pointwise, then $f$ is integrable too and $$\lim_k\int_{\mathbb{R}^n}f_k \ d\lambda = \int_{\mathbb{R}^n}f\ d\lambda.$$
We define a (postive) function to be integrable if $$\int_{\mathbb{R}^n}f\ d\lambda<\infty.$$
$\textbf{Question}$: Suppose I am integrating over some subset $A\subset\mathbb{R}^n$. And suppose there is a sequence $f_k:A\to \mathbb{C}$ such that for all $x\in A$ and all $k$ we have $|f_k(x)|\leq g(x)$ for some integrable function $g:A\to \mathbb{C}$.
$\textbf{But}$: suppose $g$ is $\textit{only}$ integrable on $A$, and not on $\mathbb{R}^n$. That is
$$\int_A g\ d\lambda<\infty \ \text{but $\textbf{not}$} \ \int_{\mathbb{R}^n}g\ d\lambda < \infty.$$
Does the dominated convergence theorem still hold here?
Yes, assuming that $A$ is a measurable subset of $\mathbb R^{n}$. Just define $f_k(x)=g(x)=0$ for $x \notin A$ and apply DCT.