# Bochner integral

An integral of a function with values in a Banach space with respect to a scalar-valued measure. It belongs to the family of so-called strong integrals.

Let $\mathcal{F}(X;E,\mathfrak{B},\mu)$ denote the vector space (over $\mathbb{R}$ or $\mathbb{C}$) of functions $f: E \to X$, where:

• $X$ is a Banach space (resp. real or complex).
• $(E,\mathfrak{B},\mu)$ is a measure space, with $\mu$ a $\sigma$-additive scalar-valued measure on a $\sigma$-algebra $\mathfrak{B}$ on $E$.

A function $s \in \mathcal{F}$ is called simple if and only if for some $n \in \mathbb{N}$, there exist distinct vectors $x_{1},\ldots,x_{n} \in X$ and pairwise-disjoint $\mathfrak{B}$-measurable subsets $B_{1},\ldots,B_{n}$ of $E$, each with finite $\mu$-measure, such that $\displaystyle s(t) = \sum_{i = 1}^{n} {\chi_{B_{i}}}(t) \cdot x_{i}$ for every $t \in E$, in which case we define the Bochner-integral of $s$ by $$\int_{E} s ~ \mathrm{d}{\mu} \stackrel{\text{df}}{=} \sum_{i = 1}^{n} \mu(B_{i}) \cdot x_{i}.$$

A function $f \in \mathcal{F}$ is called strongly measurable if and only if there exists a sequence $(s_{n})_{n \in \mathbb{N}}$ of simple functions such that $\displaystyle \lim_{n \to \infty} \| f(\bullet) - {s_{n}}(\bullet) \|_{X} = 0$ pointwise almost-everywhere on $E$ (we call this an approximating sequence for $f$), in which case the scalar-valued function $\| f(\bullet) \|_{X}: E \to [0,\infty)$ is $\mathfrak{B}$-measurable.

A function $f \in \mathcal{F}$ is called Bochner-integrable if and only if it is strongly measurable and for some sequence $(s_{n})_{n \in \mathbb{N}}$ of simple functions, we have $$\lim_{n \to \infty} \int_{E} \| f(t) - {s_{n}}(t) \|_{X} ~ \mathrm{d}{\mu(t)} = 0,$$ in which case we define the Bochner-integral of $f$ over a $\mathfrak{B}$-measurable subset $B$ of $E$ by $$\int_{B} f ~ \mathrm{d}{\mu} \stackrel{\text{df}}{=} \lim_{n \to \infty} \int_{E} \chi_{B} \cdot s_{n} ~ \mathrm{d}{\mu}.$$ This limit is taken with respect to the norm-topology on $X$. It exists and is independent of the choice of the sequence $(s_{n})_{n \in \mathbb{N}}$.

Criterion for Bochner-integrability: For a (strongly measurable) function $f \in \mathcal{F}$ to be Bochner-integrable, it is necessary and sufficient for its pointwise-norm to be integrable, i.e., $$\int_{E} \| f(t) \|_{X} ~ \mathrm{d}{\mu(t)} < \infty.$$

The set of Bochner-integrable functions is a vector subspace $\mathcal{L}$ of $\mathcal{F}$, and the Bochner-integral is a linear operator from $\mathcal{L}$ to $X$.

Properties of Bochner-integrals:

(1) $\displaystyle \left\| \int_{B} f ~ \mathrm{d}{\mu} \right\|_{X} \leq \int_{B} \| f(t) \|_{X} ~ \mathrm{d}{\mu(t)}$ for every $f \in \mathcal{L}$ and $B \in \mathfrak{B}$.

(2) The Bochner integral for a fixed $f \in \mathcal{L}$ is a $\sigma$-additive and $\mu$-absolutely continuous set-function on $\mathfrak{B}$, i.e., $$\int_{\bigcup_{i = 1}^{\infty} B_{i}} f ~ \mathrm{d}{\mu} = \sum_{i = 1}^{\infty} \int_{B_{i}} f ~ \mathrm{d}{\mu}$$ for every sequence $(B_{i})_{i \in \mathbb{N}}$ of pairwise-disjoint $\mathfrak{B}$-measurable subsets of $E$, each with finite $\mu$-measure, and $$\forall \epsilon > 0, ~ \exists \delta > 0, ~ \forall B \in \mathfrak{B}: \quad \mu(B) < \delta \quad \Longrightarrow \quad \left\| \int_{B} f ~ \mathrm{d}{\mu} \right\|_{X} < \epsilon.$$

(3) If

• $(f_{n})_{n \in \mathbb{N}}$ is a sequence in $\mathcal{L}$ that converges pointwise almost-everywhere on $B \in \mathfrak{B}$ to some $f \in \mathcal{F}$,
• $\| {f_{n}}(\bullet) \|_{X} \leq g$ pointwise almost-everywhere on $B$ for some $\mathfrak{B}$-measurable function $g: E \to [0,\infty)$, and
• $\displaystyle \int_{B} g ~ \mathrm{d}{\mu} < \infty$,

then $f \in \mathcal{L}$ and $\displaystyle \lim_{n \to \infty} \int_{B} f_{n} ~ \mathrm{d}{\mu} = \int_{B} f ~ \mathrm{d}{\mu}$.

(4) The vector space $\mathcal{L}$ is complete with respect to the norm $\| \bullet \|_{\mathcal{L}}$ defined by $$\forall f \in \mathcal{L}: \quad \| f \|_{\mathcal{L}} \stackrel{\text{df}}{=} \int_{E} \| f(t) \|_{X} ~ \mathrm{d}{\mu(t)}.$$

(5) If $T$ is a closed linear operator from a Banach space $X$ to another $Y$, and if $f \in \mathcal{L}(X;E,\mathfrak{B},\mu)$ and $T \circ f \in \mathcal{L}(Y;E,\mathfrak{B},\mu)$, then $$\forall B \in \mathfrak{B}: \quad \int_{B} T \circ f ~ \mathrm{d}{\mu} = T \! \left( \int_{B} f ~ \mathrm{d}{\mu} \right).$$ If $T$ is bounded, then the condition $T \circ f \in \mathcal{L}(Y;E,\mathfrak{B},\mu)$ is automatically fulfilled ([3][5]).

The Bochner-integral was introduced by S. Bochner in [1]. Equivalent definitions were given by T. Hildebrandt in [2] and by N. Dunford (the $D_{0}$-integral).

#### References

 [1] S. Bochner, “Integration von Funktionen, deren Werte die Elemente eines Vektorraumes sind”, Fund. Math., 20 (1933), pp. 262–276. [2] T.H. Hildebrandt, “Integration in abstract spaces”, Bull. Amer. Math. Soc., 59 (1953), pp. 111–139. [3] K. Yosida, “Functional analysis”, Springer (1980), Ch. 8, §1. [4] E. Hille and R.S. Phillips, “Functional analysis and semi-groups”, Amer. Math. Soc. (1957). [5] N. Dunford and J.T. Schwartz, “Linear operators. General theory”, 1, Interscience (1958).