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
Famous quotes containing the word definition:
“Mothers often are too easily intimidated by their childrens negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“Im beginning to think that the proper definition of Man is an animal that writes letters.”
—Lewis Carroll [Charles Lutwidge Dodgson] (18321898)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)