Bochner integral

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.

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 E_i are disjoint members of the σ-algebra Σ, the b_i are distinct elements of B, and χ_E is the characteristic function of E. If μ(E_i) is finite whenever b_i ≠ 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 s_n 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 ${\displaystyle L^{1}}$.

Many of the familiar properties of the Lebesgue integral continue to hold for the Bochner integral. Particularly useful is Bochner's criterion for integrability, which states that if (X, Σ, μ) is a measure space, then a Bochner-measurable function ƒ : X → B is Bochner integrable if and only if

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