I don't want too much of a hint, so won't display my particular example.
Given an unbounded function $\phi$. I want to compute its Lebesgue integral over $(a,b)$. It blows up to infinity at $b$ but is defined elsewhere. Being unbounded it is not Riemann integrable. The integral converges. The function is monotone increasing. I want a sequence of bounded functions that converge to $\phi$. Would $\phi_n$ the restriction of $\phi$ to $(a,b-\frac{1}{n})$ be a good choice, or does the domain of the functions in the sequence have to be $(a,b)$ if I want to use standard convergence theorems (such as dominated, monotone convergence etc)?