I'm reading Royden's Real Analysis 4ed, and have doubts about the proof of Theorem 5 in Section 8.1.
The theorem says: if $T$ is a bounded functional on $L^p[a, b]$, then there is a function $g \in L^q[a, b]$, where $q$ is the conjugate of $p$, for which $T(f) = \int_a^b g \cdot f$, $\forall f \in L^p[a, b]$.
The proof constructs such a function $g$, and then shows $T(f) = \int_a^b g \cdot f$, for all step functions in $L^p[a, b]$. Then it proceeds to show $T(f) = \int_a^b g \cdot f$, for all simple functions in $L^p[a, b]$. Finally, it shows the functional $S(f) = \int_a^b g \cdot f$, for all fucntions in $L^p[a, b]$, and then applies a previous proposition: since $T=S$ on all simple functions, and the space of simple functions is dense in $L^p[a, b]$. Therefore, $T=S$ in $L^p[a, b]$.
My question is: the space of step functions is also dense in $L^p[a, b]$(Proposition 10, Section 7.4), why bother proving $T(f) = \int_a^b g \cdot f$ for all simple functions? Can't we say, since $T=S$ on all step functions, and the space of step functions is dense in $L^p[a, b]$, hence $T=S$ on $L^p(E)$?