After a short introductory section concerning the Riemann integral of a Banach space \(X\)-valued function the concept of Riemann-Lebesgue (RL) integral [introduced by V. M. Kadets and L. M. Tseytlin in “On ‘integration’ of non-integrable vector-valued functions”, Mat. Fiz. Anal. Geom. 7, No. 1, 49–65 (2000; Zbl 0974.28007)] of such a function is presented. This integral is based on a partition of the integration domain \(\Omega\) into a countable number of measurable subsets \(\Delta_i\) and on the formal absolutely convergent “integral sum” \(\sum_i f(t_i)\mu(\Delta_i)\), \(t_i\in \Omega_i\). The RL-integral is then given by a Riemannian refinement definition based on integral sums of the above type. The set \(RL_1(\Omega, X, \mu)\) of functions integrable in this sense is a subspace of \(L_1(\Omega, X, \mu)\) (the Bochner integrable functions). A function \(g:\Omega \to X\) is said to be the Bochner integrable equivalent of \(f \in RL_1(\Omega, X, \mu)\) if \(g\in L_1(\Omega, X, \mu)\) and the respective integrals are equal.
Conditions are presented when \(f \in RL_1(\Omega, X, \mu)\) always has a Bochner integrable equivalent (e.g., \(X\) has the Radon-Nikodým property). An example shows that there is an \(f\in RL_1([0,1], L_\infty[0,1])\) which does not possess a Bochner integrable equivalent. In the final section the set \(I(f)\) of limit points of RL-integral sums is studied. Interesting results concerning the non-voidness of \(I(f)\) for a function \(f\) with \(\overline{\int}_\Omega \| f\| d\mu < \infty\) are given (\(\overline{\int}\) means the upper Lebesgue integral). For example, if \(\overline{\int}_\Omega \| f\| d\mu < \infty\) and \(I(f)\) consists of a single point \(x\in X\) then \(f\) is Pettis integrable to the value \(x\).