In special, $M_0$ in Lemma 6.26 denotes the family of all step functions on real line.
Dear friends, I wonder whether $S_j$'s are needed in the proof of Lemma 6.26. Personally, I believe that $S_0$ is enough to capture the difference between the Lebesgue integral and the Riemann sum.
The problem with using $S_0$ alone is that it is not a difference between the Lebesgue integral and the Riemann sum. In fact, $S_0<\epsilon/2$ whereas $\int \phi$ could be large. If the maximum interval size, $|P|<\delta$ were to approach 0 then $\sum\phi(t_k)\Delta x_k$ should approach $\int \phi$ even when the terms included in $S_0$ are omitted.
Although this book doesn't seem to take this approach, I prefer the following. Define Lebesgue integrals in terms up $\sup$'s and $\inf$'s of simple functions. Then demonstrate that Riemann integrability as defined in your question has a lower and upper sum formulation. Then this proof would be very short: Step functions are simple functions so $R\int_{lower}\phi\le \int_{lower}\phi \le \int_{upper}\phi\le R\int_{upper}\phi$.