ca.analysis and odes - How can we use the bounded convergence theorem in this proof of the Riesz Representation Theorem?

I'm studying the proof of the Riesz Representation Theorem as it appears in Ch. 6 of Royden's Real Analysis. When I looked on the web I noted there are a few different theorems that go by the name "Riesz Representation Theorem" so I'll state the one I'm looking at:

Let F be a bounded linear functional
on $L^p$, $1 leq p < infty$. Then
there is a function $g in L^q ni$,
$F(f) = int fg$.

The proof starts by showing that g exists for the characteristic functions $chi_s = chi_{[0,s]}$. Then we can write any step function as the sum of $chi_{s_{i}}$. So based on an earlier theorem, if f is a bounded measurable function in [0,1] we can write a sequence of step functions, $<psi_n>$ that converge to f almost everywhere. This is the sentence that confuses me:

"Since the sequence $<|f- psi_n|^p>$
is uniformly bounded and tends to zero
almost everywhere, the bounded
convergence theorem implies that
$||f-psi_n||_p rightarrow 0$."

But, when I look at the bounded convergence theorem, it would require $mathop{lim}limits_{n to infty} |f-psi_n|^p = 0$. Period. Not just almost everywhere, to get $mathop{lim}limits_{n to infty} int_{[0,1]}|f-psi_n|^p = int_{[0,1]} 0 = 0$.

So, that's where I'm stuck. I just don't see how the bounded convergence theorem can work here. (Side question: I also, don't feel I really know what Royden means by "uniformly bounded" is that just saying there is one bound that works for the whole set? How is that different from regular bounded?)