Approximation theorems for generalized Riemann integrals (Q1852389)

A function \(f\) defined on \([a,b]\) with values in a Banach space \((X,\|\cdot \|)\) is said to be integrable if for each \(\varepsilon > 0\) there exists a gauge \(\delta\) such that whenever \(\{([u_i,\nu_i],\xi_i)\}^n_{i=1}\) and \(\{([u_i,\nu_i],\xi_i')\}^n_{i=1}\) are two McShane \(\delta\)-fine divisions of \([a, b]\), we have \[ \sum^n_{i=1}\|(\nu_i-u_i)(f(\xi_i)-f(\xi_i'))\|\leq\varepsilon. \] In this note, it is proved that if \(f\) is integrable on \([a,b]\) then for each \(\varepsilon > 0\) then exists a step function \(g\) such that \(\|f - g\|_1\leq\varepsilon\), where \(\|f-g\|_1\) is the Lebesgue integral of \(\|f(t) - g(t)\|\) over \([a,b]\). Using this approximation theorem, the author gives another proof that \(f\) is integrable if and only if \(f\) is Bochner integrable and characterizes compact subsets of \(L^1\) in terms of equi-integrability for finite dimensional cases.