  The like-Lebesgue integral of real-valued measurable functions (abbreviated
as \textit{RVM-MI})is the most complete and appropriate integration Theory.
Integrals are also defined in abstract spaces since Pettis (1938). In
particular, Bochner integrals received much interest with very recent
researches. It is very commode to use the \textit{RVM-MI} in constructing
Bochner integral in Banach or in locally convex spaces. In this simple not, we
prove that the Bochner integral and the \textit{RVM-MI} with respect to a
finite measure $m$ are the same on $\mathbb{R}$. Applications of that equality
may be useful in weak limits on Banach space.
