Pettis integral

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 [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 [2].

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