View source for Pettis integral

{{TEX|done}}
An integral of a vector-valued function with respect to a scalar measure, which is a so-called weak integral. It was introduced by B.J. Pettis [[#References|[1]]].

Let $F(X,E,\mathfrak B,\mu)$ be the vector space of functions $x(t)$, $t\in E$, with values in the Banach space $X$ and given on a set $(E,\mathfrak B,\mu)$ with a countably-additive measure $\mu$ on the $\sigma$-algebra $\mathfrak B$ of subsets of $E$. The function $x(t)$ is called weakly measurable if for any $f\in X^*$ the scalar function $f[x(t)]$ is measurable. The function $x(t)$ is Pettis integrable over a measurable subset $M\subset E$ if for any $f\in X^*$ the function $f[x(t)]$ is integrable on $M$ and if there exists an element $x(M)\in X$ such that

$$f[x(M)]=\int\limits_Mf[x(t)]d\mu.$$

Then, by definition,

$$\int\limits_Mx(t)d\mu=x(M)$$

is called the Pettis integral. That integral was introduced for the case $E=(a,b)$ with the ordinary Lebesgue measure by I.M. Gel'fand [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.J. Pettis,   "On integration in vector spaces"  ''Trans. Amer. Math. Soc.'' , '''44''' :  2  (1938)  pp. 277–304</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Gel'fand,   "Sur un lemme de la théorie des espaces linéaires"  ''Zap. Naukovodosl. Inst. Mat. Mekh. Kharkov. Mat. Tov.'' , '''13''' :  1  (1936)  pp. 35–40</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  T. Hildebrandt,   "Integration in abstract spaces"  ''Bull. Amer. Math. Soc.'' , '''59'''  (1953)  pp. 111–139</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></table>



====Comments====


====References====
<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">[a2]</TD> <TD valign="top">  M. Talagrand,   "Pettis integral and measure theory" , ''Mem. Amer. Math. Soc.'' , '''307''' , Amer. Math. Soc.  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Bichteler,   "Integration theory (with special attention to vector measures)" , ''Lect. notes in math.'' , '''315''' , Springer  (1973)</TD></TR></table>