Bochner Integral - Definition

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 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 .