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Pettis integral
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<h2 id="definition">Definition</h2> <p>Let f : X → V {\displaystyle f:X\to V} where ( X , Σ , μ ) {\displaystyle (X,\Sigma ,\mu )} is a measure space and V {\displaystyle V} is a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> (TVS) with a continuous dual space V ′ {\displaystyle V'} that separates points (that is, if x ∈ V {\displaystyle x\in V} is nonzero then there is some l ∈ V ′ {\displaystyle l\in V'} such that l ( x ) ≠ 0 {\displaystyle l(x)\neq 0} ), for example, V {\displaystyle V} is a <a href="/facts/Normed_space/rmDljfyE">normed space</a> or (more generally) is a Hausdorff <a href="/facts/Locally_convex/AGrPrayC">locally convex</a> TVS. Evaluation of a functional may be written as a <a href="/facts/Dual_system/2Y8KyofV">duality pairing</a>: ⟨ φ , x ⟩ = φ [ x ] . {\displaystyle \langle \varphi ,x\rangle =\varphi [x].} </p><p>The map f : X → V {\displaystyle f:X\to V} is called weakly measurable if for all φ ∈ V ′ , {\displaystyle \varphi \in V',} the scalar-valued map φ ∘ f {\displaystyle \varphi \circ f} is a <a href="/facts/Measurable_map/NgIHnb1b">measurable map</a>. A weakly measurable map f : X → V {\displaystyle f:X\to V} is said to be weakly integrable on X {\displaystyle X} if there exists some e ∈ V {\displaystyle e\in V} such that for all φ ∈ V ′ , {\displaystyle \varphi \in V',} the scalar-valued map φ ∘ f {\displaystyle \varphi \circ f} is <a href="/facts/Lebesgue_integrable/f4AgvC6x">Lebesgue integrable</a> (that is, φ ∘ f ∈ L 1 ( X , Σ , μ ) {\displaystyle \varphi \circ f\in L^{1}\left(X,\Sigma ,\mu \right)} ) and φ ( e ) = ∫ X φ ( f ( x ) ) d μ ( x ) . {\displaystyle \varphi (e)=\int _{X}\varphi (f(x))\,\mathrm {d} \mu (x).} </p><p>The map f : X → V {\displaystyle f:X\to V} is said to be Pettis integrable if φ ∘ f ∈ L 1 ( X , Σ , μ ) {\displaystyle \varphi \circ f\in L^{1}\left(X,\Sigma ,\mu \right)} for all φ ∈ V ′ {\displaystyle \varphi \in V^{\prime }} and also for every A ∈ Σ {\displaystyle A\in \Sigma } there exists a vector e A ∈ V {\displaystyle e_{A}\in V} such that ⟨ φ , e A ⟩ = ∫ A ⟨ φ , f ( x ) ⟩ d μ ( x ) for all φ ∈ V ′ . {\displaystyle \langle \varphi ,e_{A}\rangle =\int _{A}\langle \varphi ,f(x)\rangle \,\mathrm {d} \mu (x)\quad {\text{ for all }}\varphi \in V'.} </p><p>In this case, e A {\displaystyle e_{A}} is called the Pettis integral of f {\displaystyle f} on A . {\displaystyle A.} Common notations for the Pettis integral e A {\displaystyle e_{A}} include ∫ A f d μ , ∫ A f ( x ) d μ ( x ) , and, in case that A = X is understood, μ [ f ] . {\displaystyle \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].} </p><p>To understand the motivation behind the definition of "weakly integrable", consider the special case where V {\displaystyle V} is the underlying scalar field; that is, where V = R {\displaystyle V=\mathbb {R} } or V = C . {\displaystyle V=\mathbb {C} .} In this case, every linear functional φ {\displaystyle \varphi } on V {\displaystyle V} is of the form φ ( y ) = s y {\displaystyle \varphi (y)=sy} for some scalar s ∈ V {\displaystyle s\in V} (that is, φ {\displaystyle \varphi } is just scalar multiplication by a constant), the condition φ ( e ) = ∫ A φ ( f ( x ) ) d μ ( x ) for all φ ∈ V ′ , {\displaystyle \varphi (e)=\int _{A}\varphi (f(x))\,\mathrm {d} \mu (x)\quad {\text{for all}}~\varphi \in V',} simplifies to s e = ∫ A s f ( x ) d μ ( x ) for all scalars s . {\displaystyle se=\int _{A}sf(x)\,\mathrm {d} \mu (x)\quad {\text{for all scalars}}~s.} In particular, in this special case, f {\displaystyle f} is weakly integrable on X {\displaystyle X} if and only if f {\displaystyle f} is Lebesgue integrable. </p> <h2 id="relation-to-dunford-integral">Relation to Dunford integral</h2> <p>The map f : X → V {\displaystyle f:X\to V} is said to be Dunford integrable if φ ∘ f ∈ L 1 ( X , Σ , μ ) {\displaystyle \varphi \circ f\in L^{1}\left(X,\Sigma ,\mu \right)} for all φ ∈ V ′ {\displaystyle \varphi \in V^{\prime }} and also for every A ∈ Σ {\displaystyle A\in \Sigma } there exists a vector d A ∈ V ″ , {\displaystyle d_{A}\in V'',} called the Dunford integral of f {\displaystyle f} on A , {\displaystyle A,} such that ⟨ d A , φ ⟩ = ∫ A ⟨ φ , f ( x ) ⟩ d μ ( x ) for all φ ∈ V ′ {\displaystyle \langle d_{A},\varphi \rangle =\int _{A}\langle \varphi ,f(x)\rangle \,\mathrm {d} \mu (x)\quad {\text{ for all }}\varphi \in V'} where ⟨ d A , φ ⟩ = d A ( φ ) . {\displaystyle \langle d_{A},\varphi \rangle =d_{A}(\varphi ).} </p><p>Identify every vector x ∈ V {\displaystyle x\in V} with the map scalar-valued functional on V ′ {\displaystyle V'} defined by φ ∈ V ′ ↦ φ ( x ) . {\displaystyle \varphi \in V'\mapsto \varphi (x).} This assignment induces a map called the canonical evaluation map and through it, V {\displaystyle V} is identified as a vector subspace of the double dual V ″ . {\displaystyle V''.} The space V {\displaystyle V} is a <a href="/facts/Semi-reflexive_space/NRcUlayR">semi-reflexive space</a> if and only if this map is <a href="/facts/Surjection/KezhLpCb">surjective</a>. The f : X → V {\displaystyle f:X\to V} is Pettis integrable if and only if d A ∈ V {\displaystyle d_{A}\in V} for every A ∈ Σ . {\displaystyle A\in \Sigma .} </p> <h2 id="properties">Properties</h2> <p>An immediate consequence of the definition is that Pettis integrals are compatible with continuous linear operators: If Φ : V 1 → V 2 {\displaystyle \Phi \colon V_{1}\to V_{2}} is linear and continuous and f : X → V 1 {\displaystyle f\colon X\to V_{1}} is Pettis integrable, then Φ ∘ f {\displaystyle \Phi \circ f} is Pettis integrable as well and ∫ X Φ ( f ( x ) ) d μ ( x ) = Φ ( ∫ X f ( x ) d μ ( x ) ) . {\displaystyle \int _{X}\Phi (f(x))\,d\mu (x)=\Phi \left(\int _{X}f(x)\,d\mu (x)\right).} </p><p>The standard estimate | ∫ X f ( x ) d μ ( x ) | ≤ ∫ X | f ( x ) | d μ ( x ) {\displaystyle \left|\int _{X}f(x)\,d\mu (x)\right|\leq \int _{X}|f(x)|\,d\mu (x)} for real- and complex-valued functions generalises to Pettis integrals in the following sense: For all continuous seminorms p : V → R {\displaystyle p\colon V\to \mathbb {R} } and all Pettis integrable f : X → V {\displaystyle f\colon X\to V} , p ( ∫ X f ( x ) d μ ( x ) ) ≤ ∫ X _ p ( f ( x ) ) d μ ( x ) {\displaystyle p\left(\int _{X}f(x)\,d\mu (x)\right)\leq {\underline {\int _{X}}}p(f(x))\,d\mu (x)} holds. The right-hand side is the lower Lebesgue integral of a [ 0 , ∞ ] {\displaystyle [0,\infty ]} -valued function, that is, ∫ X _ g d μ := sup { ∫ X h d μ | h : X → [ 0 , ∞ ] is measurable and 0 ≤ h ≤ g } . {\displaystyle {\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\}.} Taking a lower Lebesgue integral is necessary because the integrand p ∘ f {\displaystyle p\circ f} may not be measurable. This follows from the <a href="/facts/Hahn-Banach_theorem/3AUHusvn">Hahn-Banach theorem</a> because for every vector v ∈ V {\displaystyle v\in V} there must be a continuous functional φ ∈ V ∗ {\displaystyle \varphi \in V^{*}} such that φ ( v ) = p ( v ) {\displaystyle \varphi (v)=p(v)} and for all w ∈ V {\displaystyle w\in V} , | φ ( w ) | ≤ p ( w ) {\displaystyle |\varphi (w)|\leq p(w)} . Applying this to v := ∫ X f d μ {\displaystyle v:=\int _{X}f\,d\mu } gives the result. </p> <h3>Mean value theorem</h3> <p>An important property is that the Pettis integral with respect to a finite measure is contained in the closure of the <a href="/facts/Convex_hull/D4Id3JDS">convex hull</a> of the values scaled by the measure of the integration domain: μ ( A ) < ∞ implies ∫ A f d μ ∈ μ ( A ) ⋅ co ( f ( A ) ) ¯ {\displaystyle \mu (A)<\infty {\text{ implies }}\int _{A}f\,d\mu \in \mu (A)\cdot {\overline {\operatorname {co} (f(A))}}} </p><p>This is a consequence of the <a href="/facts/Hahn-Banach_theorem/3AUHusvn">Hahn-Banach theorem</a> and generalizes the <a href="/facts/Mean_value_theorem/JnjkzQb4">mean value theorem for integrals of real-valued functions</a>: If V = R {\displaystyle V=\mathbb {R} } , then closed convex sets are simply intervals and for f : X → [ a , b ] {\displaystyle f\colon X\to [a,b]} , the following inequalities hold: μ ( A ) a ≤ ∫ A f d μ ≤ μ ( A ) b . {\displaystyle \mu (A)a~\leq ~\int _{A}f\,d\mu ~\leq ~\mu (A)b.} </p> <h3>Existence</h3> <p>If V = R n {\displaystyle V=\mathbb {R} ^{n}} is finite-dimensional then f {\displaystyle f} is Pettis integrable if and only if each of f {\displaystyle f} ’s coordinates is Lebesgue integrable. </p><p>If f {\displaystyle f} is Pettis integrable and A ∈ Σ {\displaystyle A\in \Sigma } is a measurable subset of X {\displaystyle X} , then by definition f | A : A → V {\displaystyle f_{|A}\colon A\to V} and f ⋅ 1 A : X → V {\displaystyle f\cdot 1_{A}\colon X\to V} are also Pettis integrable and ∫ A f | A d μ = ∫ X f ⋅ 1 A d μ . {\displaystyle \int _{A}f_{|A}\,d\mu =\int _{X}f\cdot 1_{A}\,d\mu .} </p><p>If X {\displaystyle X} is a topological space, Σ = B X {\displaystyle \Sigma ={\mathfrak {B}}_{X}} its <a href="/facts/Borel_set/jvSsfpkP">Borel- σ {\displaystyle \sigma } -algebra</a>, μ {\displaystyle \mu } a <a href="/facts/Borel_measure/8cYR8Ysw">Borel measure</a> that assigns finite values to compact subsets, V {\displaystyle V} is <a href="/facts/Quasi-complete_space/rQPDf5nu">quasi-complete</a> (that is, every <i>bounded</i> <a href="/facts/Cauchy_net/msmsvMTT">Cauchy net</a> converges) and if f {\displaystyle f} is continuous with compact support, then f {\displaystyle f} is Pettis integrable. More generally: If f {\displaystyle f} is weakly measurable and there exists a compact, convex C ⊆ V {\displaystyle C\subseteq V} and a null set N ⊆ X {\displaystyle N\subseteq X} such that f ( X ∖ N ) ⊆ C {\displaystyle f(X\setminus N)\subseteq C} , then f {\displaystyle f} is Pettis-integrable. </p> <h2 id="law-of-large-numbers-for-pettis-integrable-random-variables">Law of large numbers for Pettis-integrable random variables</h2> <p>Let ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} )} be a probability space, and let V {\displaystyle V} be a topological vector space with a dual space that separates points. Let v n : Ω → V {\displaystyle v_{n}:\Omega \to V} be a sequence of Pettis-integrable random variables, and write E [ v n ] {\displaystyle \operatorname {E} [v_{n}]} for the Pettis integral of v n {\displaystyle v_{n}} (over X {\displaystyle X} ). Note that E [ v n ] {\displaystyle \operatorname {E} [v_{n}]} is a (non-random) vector in V , {\displaystyle V,} and is not a scalar value. </p><p>Let v ¯ N := 1 N ∑ n = 1 N v n {\displaystyle {\bar {v}}_{N}:={\frac {1}{N}}\sum _{n=1}^{N}v_{n}} denote the sample average. By linearity, v ¯ N {\displaystyle {\bar {v}}_{N}} is Pettis integrable, and E [ v ¯ N ] = 1 N ∑ n = 1 N E [ v n ] ∈ V . {\displaystyle \operatorname {E} [{\bar {v}}_{N}]={\frac {1}{N}}\sum _{n=1}^{N}\operatorname {E} [v_{n}]\in V.} </p><p>Suppose that the partial sums 1 N ∑ n = 1 N E [ v ¯ n ] {\displaystyle {\frac {1}{N}}\sum _{n=1}^{N}\operatorname {E} [{\bar {v}}_{n}]} converge absolutely in the topology of V , {\displaystyle V,} in the sense that all rearrangements of the sum converge to a single vector λ ∈ V . {\displaystyle \lambda \in V.} The weak law of large numbers implies that ⟨ φ , E [ v ¯ N ] − λ ⟩ → 0 {\displaystyle \langle \varphi ,\operatorname {E} [{\bar {v}}_{N}]-\lambda \rangle \to 0} for every functional φ ∈ V ∗ . {\displaystyle \varphi \in V^{*}.} Consequently, E [ v ¯ N ] → λ {\displaystyle \operatorname {E} [{\bar {v}}_{N}]\to \lambda } in the <a href="/facts/Weak_topology/XJASE9Up">weak topology</a> on X . {\displaystyle X.} </p><p>Without further assumptions, it is possible that E [ v ¯ N ] {\displaystyle \operatorname {E} [{\bar {v}}_{N}]} does not converge to λ . {\displaystyle \lambda .} To get strong convergence, more assumptions are necessary. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Bochner_measurable_function/HxXFeAgt">Bochner measurable function</a></li> <li><a href="/facts/Bochner_integral/RpzcTu9n">Bochner integral</a> – Concept in mathematics</li> <li><a href="/facts/Bochner_space/tSuPabsJ">Bochner space</a> – Type of topological space</li> <li><a href="/facts/Vector_measure/pQXUI3mv">Vector measure</a></li> <li><a href="/facts/Weakly_measurable_function/cAFxsUvL">Weakly measurable function</a></li></ul> <ul><li>James K. Brooks, <i>Representations of weak and strong integrals in Banach spaces</i>, <a href="/facts/Proceedings_of_the_National_Academy_of_Sciences_of_the_United_States_of_America/w75CThX0">Proceedings of the National Academy of Sciences of the United States of America</a> 63, 1969, 266–270. <a href="http://www.pnas.org/content/63/2/266.abstract">Fulltext</a> <a href="/facts/MR_(identifier)/uP137L11">MR</a><a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0274697">0274697</a></li> <li><a href="/facts/Israel_Gelfand/TJeJouKs">Israel M. Gel'fand</a>, <i>Sur un lemme de la théorie des espaces linéaires</i>, Commun. Inst. Sci. Math. et Mecan., Univ. Kharkoff et Soc. Math. Kharkoff, IV. Ser. 13, 1936, 35–40 <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0014.16202">0014.16202</a></li> <li><a href="/facts/Michel_Talagrand/b7xTaT7f">Michel Talagrand</a>, <i>Pettis Integral and Measure Theory</i>, Memoirs of the AMS no. 307 (1984) <a href="/facts/MR_(identifier)/uP137L11">MR</a><a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0756174">0756174</a></li> <li>Sobolev, V. I. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Pettis_integral">"Pettis integral"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li></ul>