Definition
Let (X, Σ, μ) be a measure space and B a Banach space. The Bochner integral is defined in much the same way as the Lebesgue integral. First, a simple function is any finite sum of the form
where the Ei are disjoint members of the σ-algebra Σ, the bi are distinct elements of B, and χE is the characteristic function of E. If μ(Ei) is finite whenever bi ≠ 0, then the simple function is integrable, and the integral is then defined by
exactly as it is for the ordinary Lebesgue integral.
A measurable function ƒ : X → B is Bochner integrable if there exists a sequence of integrable simple functions sn such that
where the integral on the left-hand side is an ordinary Lebesgue integral.
In this case, the Bochner integral is defined by
It can be shown that a function is Bochner integrable if and only if it lies in the Bochner space .
Read more about this topic: Bochner Integral
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