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In [[mathematics]], the '''Pettis integral''' or '''Gelfand&ndash;Pettis integral''', named after [[Israel Gelfand|Israel M. Gelfand]] and [[Billy James Pettis]], extends the definition of the [[Lebesgue integral]] to vector-valued functions on a [[measure space]], by exploiting [[Dual system|duality]]. The integral was introduced by Gelfand for the case when the measure space is an interval with [[Lebesgue measure]]. The integral is also called the '''weak integral''' in contrast to the [[Bochner integral]], which is the strong integral.

==Definition==

Let <math>f : X \to V</math> where <math>(X,\Sigma,\mu)</math> is a measure space and <math>V</math> is a [[topological vector space]] (TVS) with a continuous dual space <math>V'</math> that separates points (that is, if <math>x \in V</math> is nonzero then there is some <math>l \in V'</math> such that <math>l(x) \neq 0</math>), for example, <math>V</math> is a [[normed space]] or (more generally) is a Hausdorff [[locally convex]] TVS.  
Evaluation of a functional may be written as a [[Dual system|duality pairing]]: 
<math display=block>\langle \varphi, x \rangle = \varphi[x].</math>

The map <math>f : X \to V</math> is called '''{{em|{{visible anchor|weakly measurable}}}}''' if for all <math>\varphi \in V',</math> the scalar-valued map <math>\varphi \circ f</math> is a [[measurable map]]. 
A weakly measurable map <math>f : X \to V</math> is said to be '''{{em|{{visible anchor|weakly integrable}} on <math>X</math>}}''' if there exists some <math>e \in V</math> such that for all <math>\varphi \in V',</math> the scalar-valued map <math>\varphi \circ f</math> is [[Lebesgue integrable]] (that is, <math>\varphi \circ f \in L^1\left( X, \Sigma, \mu \right)</math>) and 
<math display="block">\varphi(e) =  \int_X \varphi(f(x)) \, \mathrm{d} \mu(x).</math> 

The map <math>f : X \to V</math> is said to be '''{{em|{{visible anchor|Pettis integrable}}}}''' if <math>\varphi \circ f \in L^1\left( X, \Sigma, \mu \right)</math> for all <math>\varphi \in V^{\prime}</math> and also for every <math>A \in \Sigma</math> there exists a vector <math>e_A \in V</math> such that 
<math display="block">\langle \varphi, e_A \rangle = \int_A \langle \varphi, f(x) \rangle \, \mathrm{d} \mu(x) \quad \text{ for all } \varphi \in V'.</math>

In this case, <math>e_A</math> is called the '''{{em|{{visible anchor|Pettis integral}}}}''' of <math>f</math> on <math>A.</math> Common notations for the Pettis integral <math>e_A</math> include 
<math display="block">\int_A f \, \mathrm{d}\mu, \qquad \int_A f(x) \, \mathrm{d}\mu(x), \quad \text{and, in case that}~ A=X ~ \text{is understood,} \quad \mu[f].</math>

To understand the motivation behind the definition of "weakly integrable", consider the special case where <math>V</math> is the underlying scalar field; that is, where <math>V = \R </math> or <math>V = \Complex.</math> In this case, every linear functional <math>\varphi</math> on <math>V</math> is of the form <math>\varphi(y) = s y</math> for some scalar <math>s \in V</math> (that is, <math>\varphi</math> is just scalar multiplication by a constant), the condition 
<math display="block">\varphi(e) = \int_A \varphi(f(x)) \, \mathrm{d} \mu(x) \quad\text{for all}~ \varphi \in V',</math>
simplifies to 
<math display=block>s e =  \int_A s f(x) \, \mathrm{d} \mu(x) \quad\text{for all scalars}~ s.</math>
In particular, in this special case, <math>f</math> is weakly integrable on <math>X</math> if and only if <math>f</math> is Lebesgue integrable.

==Relation to Dunford integral==

The map <math>f : X \to V</math> is said to be '''{{em|{{visible anchor|Dunford integrable}}}}''' if <math>\varphi \circ f \in L^1\left( X, \Sigma, \mu \right)</math> for all <math>\varphi \in V^{\prime}</math> and also for every <math>A \in \Sigma</math> there exists a vector <math>d_A \in V'',</math> called the '''{{em|{{visible anchor|Dunford integral}}}}''' of <math>f</math> on <math>A,</math> such that 
<math display=block>\langle d_A, \varphi \rangle = \int_A \langle \varphi, f(x) \rangle \, \mathrm{d} \mu(x) \quad \text{ for all } \varphi \in V'</math> 
where <math>\langle d_A, \varphi \rangle = d_A(\varphi).</math> 

Identify every vector <math>x \in V</math> with the map scalar-valued functional on <math>V'</math> defined by <math>\varphi \in V' \mapsto \varphi(x).</math> This assignment induces a map called the canonical evaluation map and through it, <math>V</math> is identified as a vector subspace of the double dual <math>V''.</math> 
The space <math>V</math> is a [[semi-reflexive space]] if and only if this map is [[Surjection|surjective]]. 
The <math>f : X \to V</math> is Pettis integrable if and only if <math>d_A \in V</math> for every <math>A \in \Sigma.</math>

==Properties==

An immediate consequence of the definition is that Pettis integrals are compatible with continuous linear operators: If <math>\Phi \colon V_1 \to V_2</math> is linear and continuous and <math>f \colon X \to V_1</math> is Pettis integrable, then <math>\Phi\circ f</math> is Pettis integrable as well and <math display=block>\int_X \Phi(f(x))\,d\mu(x) = \Phi \left(\int_X f(x)\,d\mu(x) \right).</math>

The standard estimate <math display=block> \left |\int_X f(x)\,d\mu(x) \right | \leq \int_X |f(x)| \, d\mu(x)</math> for real- and complex-valued functions generalises to Pettis integrals in the following sense: For all continuous seminorms <math>p\colon V\to\mathbb{R}</math> and all Pettis integrable <math>f \colon X \to V</math>, <math display=block>p \left (\int_X f(x)\,d\mu(x) \right ) \leq  \underline{\int_X} p(f(x)) \,d\mu(x)</math> holds. The right-hand side is the lower Lebesgue integral of a <math>[0,\infty]</math>-valued function, that is, <math display=block> \underline{\int_X} g \,d\mu := \sup \left \{ \left. \int_X h\,d\mu \; \right  | \; h \colon X \to [0,\infty] \text{ is measurable and } 0 \leq h \leq g \right \}.</math> Taking a lower Lebesgue integral is necessary because the integrand <math>p\circ f</math> may not be measurable. This follows from the [[Hahn-Banach theorem]] because for every vector <math>v\in V</math> there must be a continuous functional <math>\varphi\in V^*</math> such that <math>\varphi(v) = p(v)</math> and for all <math>w \in V</math>, <math>|\varphi(w)|\leq p(w)</math>. Applying this to <math>v := \int_X f \, d\mu</math> gives the result.

===Mean value theorem===

An important property is that the Pettis integral with respect to a finite measure is contained in the closure of the [[convex hull]] of the values scaled by the measure of the integration domain:
<math display=block>\mu(A) < \infty \text{ implies } \int_A f\,d\mu \in \mu(A) \cdot \overline{\operatorname{co}(f(A))}</math>

This is a consequence of the [[Hahn-Banach theorem]] and generalizes the [[Mean value theorem#Mean value theorems for definite integrals|mean value theorem for integrals of real-valued functions]]: If <math>V = \R</math>, then closed convex sets are simply intervals and for <math>f \colon X \to [a, b]</math>, the following inequalities hold:
<math display=block>\mu(A) a ~\leq~ \int_A f \, d\mu ~\leq~ \mu(A)b.</math>

===Existence===

If <math>V = \R^n</math> is finite-dimensional then <math>f</math> is Pettis integrable if and only if each of <math>f</math>’s coordinates is Lebesgue integrable.

If <math>f</math> is Pettis integrable and <math>A\in\Sigma</math> is a measurable subset of <math>X</math>, then by definition <math>f_{|A} \colon A\to V</math> and <math>f \cdot 1_A \colon X \to V</math> are also Pettis integrable and <math display=block>\int_A f_{|A} \,d\mu = \int_X f \cdot 1_A \,d\mu.</math>

If <math>X</math> is a topological space, <math>\Sigma = \mathfrak{B}_X </math> its [[Borel set|Borel-<math>\sigma</math>-algebra]], <math>\mu</math> a [[Borel measure]] that assigns finite values to compact subsets, <math>V</math> is [[Quasi-complete space|quasi-complete]] (that is, every ''bounded'' [[Cauchy net]] converges) and if <math>f</math> is continuous with compact support, then <math>f</math> is Pettis integrable. 
More generally: If <math>f</math> is weakly measurable and there exists a compact, convex <math>C\subseteq V</math> and a null set <math>N\subseteq X</math> such that <math>f(X \setminus N) \subseteq C</math>, then <math>f</math> is Pettis-integrable.

==Law of large numbers for Pettis-integrable random variables==

Let <math>(\Omega, \mathcal F, \operatorname P)</math> be a probability space, and let <math>V</math> be a topological vector space with a dual space that separates points. Let <math>v_n : \Omega \to V</math> be a sequence of Pettis-integrable random variables, and write <math>\operatorname E[v_n]</math> for the Pettis integral of <math>v_n</math> (over <math>X</math>). Note that <math>\operatorname E[v_n]</math> is a (non-random) vector in <math>V,</math> and is not a scalar value.

Let 
<math display=block>\bar v_N := \frac{1}{N} \sum_{n=1}^N v_n</math> 
denote the sample average. By linearity, <math>\bar v_N</math> is Pettis integrable, and 
<math display=block>\operatorname E[\bar v_N] = \frac{1}{N} \sum_{n=1}^N \operatorname E[v_n] \in V.</math>

Suppose that the partial sums 
<math display=block>\frac{1}{N} \sum_{n=1}^N \operatorname E[\bar v_n]</math> 
converge absolutely in the topology of <math>V,</math> in the sense that all rearrangements of the sum converge to a single vector <math>\lambda \in V.</math> The weak law of large numbers implies that <math>\langle \varphi, \operatorname E[\bar v_N] - \lambda \rangle \to 0</math> for every functional <math>\varphi \in V^*.</math> Consequently, <math>\operatorname E[\bar v_N] \to \lambda</math> in the [[weak topology]] on <math>X.</math>

Without further assumptions, it is possible that <math>\operatorname E[\bar v_N]</math> does not converge to <math>\lambda.</math>{{Citation needed|date=October 2013}} To get strong convergence, more assumptions are necessary.{{Citation needed|date=October 2013}}

==See also==

* {{annotated link|Bochner measurable function}}
* {{annotated link|Bochner integral}}
* {{annotated link|Bochner space}}
* {{annotated link|Vector measure}}
* {{annotated link|Weakly measurable function}}

==References==

{{reflist}}
{{reflist|group=note}}

* James K. Brooks, ''Representations of weak and strong integrals in Banach spaces'', [[Proceedings of the National Academy of Sciences of the United States of America]] 63,  1969, 266–270. [http://www.pnas.org/content/63/2/266.abstract Fulltext] {{MR|0274697}}
* [[Israel Gelfand|Israel M. Gel'fand]], ''Sur un lemme de la théorie des espaces linéaires'', Commun. Inst. Sci. Math. et Mecan., Univ. Kharkoff et Soc. Math. Kharkoff, IV. Ser. 13, 1936, 35–40 {{Zbl|0014.16202}}
* [[Michel Talagrand]], ''Pettis Integral and Measure Theory'', Memoirs of the AMS no. 307 (1984) {{MR|0756174}}
* {{springer|title=Pettis integral|id=p/p072490|first=V. I.|last=Sobolev}}

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{{Analysis in topological vector spaces}}
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[[Category:Functional analysis]]
[[Category:Integrals]]