I'm trying to understand theorem 5 in Royden's Real Analysis, and don't quite understand some part of using Lebesgue dominated convergence theorem.
Theorem 5: Suppose $\mathit T\,$ is a bounded linear functional on $\mathit L^P[a,b]$.Then there is a function $\mathcal g\,$ in $\mathit L^q[a,b]\,$, where $\mathcal q\,$ is the congjugate of $\mathcal p\,$, for which $\mathit T(f)$=$\int_a^b$$\mathcal g\cdot\mathcal f$ for all $\mathcal f\,$ in $\mathit L^P[a,b]$.
The proof shows $\mathit T(f)$=$\int_a^b$$\mathcal g\cdot\mathcal f$ for all step functions $\mathcal f\,$ in $\mathit L^P[a,b]$ .Then assert that the equation holds for all simple functions in $\mathit L^P[a,b]$, by using continuity of $\mathit T\,$, the density of step functions which infers from proposition 10 of the preceding chapter.
I know that the sequence of step functions {$\rho_n$} converges to a simple function $\mathcal f $
in $\mathit L^P[a,b]$.But, I don't understand why permissible to use Lebesgue Dominated Convergence Theorem to show that $\lim_{n\to\infty}\int_a^b$$\mathcal g\cdot\rho_n=\int_a^b$$\mathcal g\cdot\mathcal f$. Does $\mathcal g\cdot\rho_n$ converges pointwise to $\mathcal g\cdot\mathcal f$ and $\mathcal g\cdot\mathcal f$ dominates $\mathcal g\cdot\rho_n$ ?
Thanks in advance.