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 (cf. Strong integral).

Let $ F(X;E,\mathfrak{B},\mu) $ denote the vector space of functions $ x: E \to X $, where $ X $ is a Banach space and $ (E,\mathfrak{B},\mu) $ is a measure space, with $ \mu $ being a countably-additive scalar-valued measure on a $ \sigma $-algebra $ \mathfrak{B} $ of subsets of $ E $. A function $ x_{0} \in F $ is called simple if and only if for some $ n \in \mathbb{N} $, there exist distinct points $ x_{1},\ldots,x_{n} \in X $ and measurable subsets $ B_{1},\ldots,B_{n} \in \mathfrak{B} $, satisfying $ \mu(B_{i}) < \infty $ for all $ i \in \{ 1,\ldots,n \} $ as well as $ B_{i} \cap B_{j} = \varnothing $ for all distinct $ i,j \in \{ 1,\ldots,n \} $, such that $$ \forall t \in E: \quad {x_{0}}(t) = \begin{cases} x_{i} & \text{if $ t \in B_{i} $ for some $ i \in \{ 1,\ldots,n \} $}; \\ 0 & \text{if $ \displaystyle t \in E \Bigg\backslash \bigcup_{i = 1}^{n} B_{i} $}. \end{cases} $$

A function $ x \in F $ is called strongly measurable if and only if there exists a sequence $ (x_{n})_{n \in \mathbb{N}} $ of simple functions such that $ \displaystyle \lim_{n \to \infty} \| {x_{n}}(\bullet) - x(\bullet) \|_{X} = 0 $ pointwise almost-everywhere on $ E $ with respect to $ \mu $. In this case, the scalar-valued function $ \| x(\bullet) \|_{X}: E \to [0,\infty) $ is $ \mathfrak{B} $-measurable. For the simple function $ x_{0} $, define its Bochner-integral by $$ \int_{E} {x_{0}}(t) ~ \mathrm{d}{\mu(t)} \stackrel{\text{df}}{=} \sum_{i = 1}^{n} \mu(B_{i}) \cdot x_{i}. $$

A function $ x \in F $ is called Bochner-integrable if and only if it is strongly measurable and for some approximating sequence $ (x_{n})_{n \in \mathbb{N}} $ of simple functions, we have $$ \lim_{n \to \infty} \int_{E} \| x(t) - {x_{n}}(t) \|_{X} ~ \mathrm{d}{\mu(t)} = 0. $$ The Bochner-integral of such a function over a measurable subset $ B \in \mathfrak{B} $ is then defined by $$ \int_{B} x(t) ~ \mathrm{d}{\mu(t)} \stackrel{\text{df}}{=} \lim_{n \to \infty} \int_{E} {\chi_{B}}(t) \cdot {x_{n}}(t) ~ \mathrm{d}{\mu(t)}, $$ where $ \chi_{B} $ denotes the characteristic function of $ B $, and the limit is understood in the sense of strong convergence in $ X $. This limit exists and is independent of the choice of an approximating sequence of simple functions.

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

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

Properties of Bochner-integrals:

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

2) A Bochner integral is a countably-additive $ \mu $-absolutely continuous set-function on the $ \sigma $-algebra $ \mathfrak{B} $, i.e., $$ \int_{\bigcup_{i = 1}^{\infty} B_{i}} x(t) ~ \mathrm{d}{\mu(t)} = \sum_{i = 1}^{\infty} \int_{B_{i}} x(t) ~ \mathrm{d}{\mu(t)}, $$ if $ B_{i} \in \mathfrak{B} $ and $ \mu(B_{i}) < \infty $ for all $ i \in \{ 1,\ldots,n \} $, $ B_{i} \cap B_{j} = \varnothing $ for all distinct $ i,j \in \{ 1,\ldots,n \} $, and $ \displaystyle \left\| \int_{B} x(t) ~ \mathrm{d}{\mu(t)} \right\|_{X} \to 0 $ if $ \mu(B) \to 0 $, uniformly for $ B \in \mathfrak{B} $.

3) If $ (x_{n})_{n \in \mathbb{N}} $ is a sequence in $ F $ and $ \displaystyle \lim_{n \to \infty} x_{n} = x $ pointwise almost-everywhere on $ B \in \mathfrak{B} $ with respect to $ \mu $, if $ \| {x_{n}}(\bullet) \|_{X} \leq f $ pointwise almost-everywhere on $ B $ with respect to $ \mu $, and if $ \displaystyle \int_{B} f ~ \mathrm{d}{\mu} < \infty $, then $$ x \in \mathcal{L} \qquad \text{and} \qquad \lim_{n \to \infty} \int_{B} {x_{n}}(t) ~ \mathrm{d}{\mu(t)} = \int_{B} x(t) ~ \mathrm{d}{\mu(t)}. $$

4) The space $ \mathcal{L} $ is complete with respect to the norm (cf. Convergence in norm) $$ \| x - y \|_{\mathcal{L}} \stackrel{\text{df}}{=} \int_{E} \| x(t) - y(t) \|_{X} ~ \mathrm{d}{\mu(t)}. $$

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

The Bochner integral was introduced by S. Bochner [1]. Equivalent definitions were given by T. Hildebrandt [2] and 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, R.S. Phillips, “Functional analysis and semi-groups”, Amer. Math. Soc. (1957).
[5]	N. Dunford, J.T. Schwartz, “Linear operators. General theory”, 1, Interscience (1958).

Comments

A simple function is also called a step function. A good recent textbook on integrals with values in a Banach space is [a1]; [a4] is specifically about the Bochner-integral.

References

[a1]	J. Diestel, J.J. Uhl, Jr., “Vector measures”, Math. Surveys, 15, Amer. Math. Soc. (1977).
[a2]	A.C. Zaanen, “Integration”, North-Holland (1967).
[a3]	N. Bourbaki, “Elements of mathematics. Integration”, Addison-Wesley (1975), Ch. 6, 7, 8 (translated from French).
[a4]	J. Mikusiński, “The Bochner integral”, Acad. Press (1978).