It exists and is independent of the choice of the sequence. In mathematics the bochner integral named for salomon bochner extends the definition of lebesgue integral to functions that take values in a banach space as the limit of integrals of simple functions.
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Definition of the bochner integral.
The Bochner Integral Google Drive. 3 the bochner integral in this section we will describe a construction o f the bochner integral on x with respect to the ex tension of µ to a complete σ additive measure on x. A function is called bochner integrable if and only if it is strongly measurable and for some sequence of simple functions we have in which case we define the bochner integral of over a measurable subset of by this limit is taken with respect to the norm topology on. X be a banach space bx the borel σ field of x and ωfµ a measure space with finite measure µ.
In mathematics the bochner integral named for salomon bochner extends the definition of lebesgue integral to functions that take values in a banach space as the limit of integrals of simple functions. Properties of the bochner integral we collect some further basic properties of the bochner integral that are analogoustothelebesgueintegral. This chapter is a slight modification of chap.
The bochner integral of a function ƒ with suitable properties from a measurable set a to a barach space b is the limit of the integrals over a of a sequence of simple functions s n from a to b such that the limit of the integral over a of the norm of ƒ s n approaches zero. Let aa 1a 2 b d such that a a 1a 2 and suppose that.
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