# The Real-Valued Bochner integral and the Modern Real-Valued Measurable function on $\mathbb{R}$

## TL;DR

This paper demonstrates that the real-valued Bochner integral and the RVM-MI are equivalent on the real line, simplifying the understanding of integration in Banach spaces and aiding in weak limit analysis.

## Contribution

It establishes the equivalence of Bochner and RVM-MI integrals on , providing a unified approach for integration in real and Banach spaces.

## Key findings

- Bochner integral equals RVM-MI on .
- Simplifies the use of RVM-MI in constructing Bochner integrals.
- Facilitates analysis of weak limits in Banach spaces.

## Abstract

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.