In Apostol calculus Part 1 page 74 integrabilty of a bounded function on a closed interval [a,b] is defined as follows
Let $S_f=\{s:[a,b]\rightarrow\mathbb{R}: s\ \text{is a step function and } s\leq f\},\\ T_f=\{t:[a,b]\rightarrow\mathbb{R}: t\ \text{is a step function and } t\geq f\}.$
Let $f:[a,b]\rightarrow\mathbb{R}$ be a bounded function. The lower integral of $f$ is the quantity $ \underline{I}(f) =\sup\left\{\int_a^bs(x)dx:s\in S_f\right\},$
and the upper integral of $f$ is the quantity $\overline{I}(f)=\inf \left\{\int_a^bt(x)dx:t\in T_f\right\}.$ We say that $f$ is Riemann integrable if $\underline{I}(f)=\overline{I}(f)$, in which case this quantity is called the integral of $f$ over $[a,b]$ and denoted by $\int_a^bf(x)dx$.
I wanted to know as to how this definition of Riemann integrability equivalent to the Darboux Integral definition?