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).

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 and 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).