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m (Made improvements to the clarity of the exposition.)
 
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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|Strong 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 integral|''strong integrals'']].
  
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
+
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
 
$$
 
$$
\forall t \in E: \quad
+
\int_{E} s ~ \mathrm{d}{\mu} \stackrel{\text{df}}{=} \sum_{i = 1}^{n} \mu(B_{i}) \cdot x_{i}.
{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
+
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.
$$
 
\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
+
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} \| x(t) - {x_{n}}(t) \|_{X} ~ \mathrm{d}{\mu(t)} = 0.
+
\lim_{n \to \infty} \int_{E} \| f(t) - {s_{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
+
in which case we define the Bochner-integral of $ f $ over a $ \mathfrak{B} $-measurable subset $ B $ of $ E $ 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)},
+
\int_{B} f ~ \mathrm{d}{\mu} \stackrel{\text{df}}{=} \lim_{n \to \infty} \int_{E} \chi_{B} \cdot s_{n} ~ \mathrm{d}{\mu}.
 
$$
 
$$
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.
+
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 $ x \in F $ to be Bochner-integrable, it is necessary and sufficient for its pointwise-norm to be integrable, i.e.,
+
'''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} \| x(t) \|_{X} ~ \mathrm{d}{\mu(t)} < \infty.
+
\int_{E} \| f(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.
+
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:'''
 
'''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} $.
+
(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) A Bochner integral is a countably-additive $ \mu $-absolutely continuous set-function on the $ \sigma $-algebra $ \mathfrak{B} $<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016710/b01671036.png" />, i.e.,
+
(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}} x(t) ~ \mathrm{d}{\mu(t)} = \sum_{i = 1}^{\infty} \int_{B_{i}} x(t) ~ \mathrm{d}{\mu(t)},
+
\int_{\bigcup_{i = 1}^{\infty} B_{i}} f ~ \mathrm{d}{\mu} = \sum_{i = 1}^{\infty} \int_{B_{i}} f ~ \mathrm{d}{\mu}
 
$$
 
$$
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} $.
+
for every sequence $ (B_{i})_{i \in \mathbb{N}} $ of pairwise-disjoint $ \mathfrak{B} $-measurable subsets of $ E $, each with finite $ \mu $-measure, and
 
 
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
+
\forall \epsilon > 0, ~ \exists \delta > 0, ~ \forall B \in \mathfrak{B}: \quad
\lim_{n \to \infty} \int_{B} {x_{n}}(t) ~ \mathrm{d}{\mu(t)} = \int_{B} x(t) ~ \mathrm{d}{\mu(t)}.
+
\mu(B) < \delta \quad \Longrightarrow \quad \left\| \int_{B} f ~ \mathrm{d}{\mu} \right\|_{X} < \epsilon.
 
$$
 
$$
  
4) The space $ \mathcal{L} $ is complete with respect to the norm (cf. [[Convergence in norm|Convergence in norm]])
+
(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 [[Convergence in norm|complete]] with respect to the norm $ \| \bullet \|_{\mathcal{L}} $ defined by
 
$$
 
$$
\| x - y \|_{\mathcal{L}} \stackrel{\text{df}}{=} \int_{E} \| x(t) - y(t) \|_{X} ~ \mathrm{d}{\mu(t)}.
+
\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
+
(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
$$
 
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
 
\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).
+
\int_{B} T \circ f ~ \mathrm{d}{\mu} = T \! \left( \int_{B} f ~ \mathrm{d}{\mu} \right).
 
$$
 
$$
If $ T $ is bounded, then the condition
+
If $ T $ is bounded, then the condition $ T \circ f \in \mathcal{L}(Y;E,\mathfrak{B},\mu) $ is automatically fulfilled ([[#References|[3]]]–[[#References|[5]]]).
$$
 
T \circ x \in \mathcal{L}(Y;E,\mathfrak{B},\mu)
 
$$
 
is automatically fulfilled ([[#References|[3]]]–[[#References|[5]]]).
 
  
The Bochner integral was introduced by S. Bochner [[#References|[1]]]. Equivalent definitions were given by T. Hildebrandt [[#References|[2]]] and N. Dunford (the $ D_{0} $-integral).
+
The Bochner-integral was introduced by S. Bochner in [[#References|[1]]]. Equivalent definitions were given by T. Hildebrandt in [[#References|[2]]] and by N. Dunford (the $ D_{0} $-integral).
  
 
====References====
 
====References====
  
 
<table>
 
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> S. Bochner, “Integration von Funktionen, deren Werte die Elemente eines Vektorraumes sind” ''Fund. Math.'', '''20''' (1933), pp. 262–276.</TD></TR>
+
<TR><TD valign="top">[1]</TD> <TD valign="top"> S. Bochner, “Integration von Funktionen, deren Werte die Elemente eines Vektorraumes sind”, ''Fund. Math.'', '''20''' (1933), pp. 262–276.</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> T.H. Hildebrandt, “Integration in abstract spaces” ''Bull. Amer. Math. Soc.'', '''59''' (1953), pp. 111–139.</TD></TR>
+
<TR><TD valign="top">[2]</TD> <TD valign="top"> T.H. Hildebrandt, “Integration in abstract spaces”, ''Bull. Amer. Math. Soc.'', '''59''' (1953), pp. 111–139.</TD></TR>
 
<TR><TD valign="top">[3]</TD> <TD valign="top"> K. Yosida, “Functional analysis”, Springer (1980), Ch. 8, §1.</TD></TR>
 
<TR><TD valign="top">[3]</TD> <TD valign="top"> K. Yosida, “Functional analysis”, Springer (1980), Ch. 8, §1.</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top"> E. Hille, R.S. Phillips, “Functional analysis and semi-groups”, Amer. Math. Soc. (1957).</TD></TR>
+
<TR><TD valign="top">[4]</TD> <TD valign="top"> E. Hille and R.S. Phillips, “Functional analysis and semi-groups”, Amer. Math. Soc. (1957).</TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top"> N. Dunford, J.T. Schwartz, “Linear operators. General theory”, '''1''', Interscience (1958).</TD></TR>
+
<TR><TD valign="top">[5]</TD> <TD valign="top"> N. Dunford and J.T. Schwartz, “Linear operators. General theory”, '''1''', Interscience (1958).</TD></TR>
 
</table>
 
</table>
  
 
====Comments====
 
====Comments====
A simple function is also called a step function. A good recent textbook on integrals with values in a Banach space is [[#References|[a1]]]; [[#References|[a4]]] is specifically about the Bochner-integral.
+
 
 +
A simple function is also called a ''step-function''. A good recent textbook on integrals with values in a Banach space is [[#References|[a1]]]; [[#References|[a4]]] is specifically about the Bochner-integral.
  
 
====References====
 
====References====
 +
 
<table>
 
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Diestel, J.J. Uhl, Jr., “Vector measures”, ''Math. Surveys'', '''15''', Amer. Math. Soc. (1977).</TD></TR>
+
<TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Diestel and J.J. Uhl, Jr., “Vector measures”, ''Math. Surveys'', '''15''', Amer. Math. Soc. (1977).</TD></TR>
 
<TR><TD valign="top">[a2]</TD> <TD valign="top"> A.C. Zaanen, “Integration”, North-Holland (1967).</TD></TR>
 
<TR><TD valign="top">[a2]</TD> <TD valign="top"> A.C. Zaanen, “Integration”, North-Holland (1967).</TD></TR>
 
<TR><TD valign="top">[a3]</TD> <TD valign="top"> N. Bourbaki, “Elements of mathematics. Integration”, Addison-Wesley (1975), Ch. 6, 7, 8 (translated from French).</TD></TR>
 
<TR><TD valign="top">[a3]</TD> <TD valign="top"> N. Bourbaki, “Elements of mathematics. Integration”, Addison-Wesley (1975), Ch. 6, 7, 8 (translated from French).</TD></TR>
 
<TR><TD valign="top">[a4]</TD> <TD valign="top"> J. Mikusiński, “The Bochner integral”, Acad. Press (1978).</TD></TR>
 
<TR><TD valign="top">[a4]</TD> <TD valign="top"> J. Mikusiński, “The Bochner integral”, Acad. Press (1978).</TD></TR>
 
</table>
 
</table>

Latest revision as of 21:30, 26 April 2016

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).
How to Cite This Entry:
Bochner integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bochner_integral&oldid=38659
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article