BochnerIntegral

Bochner Integral

Preliminaries

Let $(X, \Sigma, \mu)$ be a measure space, and let $E$ be a Banach space with norm $| \cdot |$.

Simple Functions

A function $s: X \rightarrow E$ is called a simple function if it takes only finitely many values in $E$. Formally, it can be written as:

$$s(x) = \sum_{i=1}^{n} \chi_{A_i}(x) e_i$$

where $\chi_{A_i}$ is the characteristic function of the measurable set $A_i$ (i.e., $\chi_{A_i}(x) = 1$ if $x \in A_i$ and $\chi_{A_i}(x) = 0$ otherwise), and $e_i$ are vectors in $E$.

The integral of a simple function $s$ is then defined as:

$$\int_{X} s , d\mu = \sum_{i=1}^{n} \mu(A_i) e_i$$

Bochner Integrable Functions

A function $f: X \rightarrow E$ is said to be Bochner integrable if there exists a sequence of simple functions $(s_n)$ such that:

1. $s_n$ is measurable for each $n$.
2. $\lim_{n \to \infty} s_n(x) = f(x)$ for almost every $x \in X$.
3. $\lim_{n \to \infty} \int_{X} | s_n - f | , d\mu = 0$.

Bochner Integral

If $f$ is Bochner integrable, then its Bochner integral is defined as:

$$\int_{X} f d\mu = \lim_{n \to \infty} \int_{X} s_n d\mu$$

This limit exists and is unique (independent of the choice of the approximating simple functions $s_n$) due to the properties of Banach spaces.

Properties

The Bochner integral has properties analogous to the Lebesgue integral, such as linearity, and it allows you to integrate functions with values in more general spaces than just $\mathbb{R}$ or $\mathbb{C}$. This is useful in various areas of mathematics and its applications, including functional analysis and partial differential equations.