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.
Definition.
Let ("X", Σ, μ) be a measure space and "B" a Banach space. The Bochner integral of a function formula_1 is defined in much the same way as the Lebesgue integral. First, define a simple function to be 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 χE 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 the sequence formula_6 is a Cauchy sequence in the Banach space formula_7, hence the limit on the right exists; furthermore, the limit is independent of the approximating sequence of simple functions formula_8. These remarks show that the integral is well-defined (i.e independent of any choices). It can be shown that a function is Bochner integrable if and only if it lies in the Bochner space formula_9.
Properties.
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
A function "ƒ" : "X" → "B"  is called Bochner-measurable if it is equal μ-almost everywhere to a function "g" taking values in a separable subspace "B"0 of "B", and such that the inverse image "g"−1("U") of every open set "U"  in "B"  belongs to Σ. Equivalently, "ƒ" is limit μ-almost everywhere of a sequence of simple functions.
If formula_11 is a continuous linear operator, and formula_12 is Bochner-integrable, then formula_13 is Bochner-integrable and integration and formula_11 may be interchanged:
This also holds for closed operators, given that formula_13 be itself integrable (which, via the criterion mentioned above is trivially true for bounded formula_11).
A version of the dominated convergence theorem also holds for the Bochner integral. Specifically, if "ƒ""n" : "X" → "B" is a sequence of measurable functions on a complete measure space tending almost everywhere to a limit function "ƒ", and if
for almost every "x" ∈ "X", and "g" ∈ "L"1(μ), then
as "n" → ∞ and
for all "E" ∈ Σ.
If "ƒ" is Bochner integrable, then the inequality
holds for all "E" ∈ Σ. In particular, the set function
defines a countably-additive "B"-valued vector measure on "X" which is absolutely continuous with respect to μ.
Radon–Nikodym property.
An important fact about the Bochner integral is that the Radon–Nikodym theorem "fails" to hold in general. This results in an important property of Banach spaces known as the Radon–Nikodym property. Specifically, if μ is a measure on ("X", Σ), then "B" has the Radon–Nikodym property with respect to μ if, for every countably-additive vector measure formula_23 on ("X", Σ) with values in "B" which has bounded variation and is absolutely continuous with respect to μ, there is a μ-integrable function "g" : "X" → "B" such that
for every measurable set "E" ∈ Σ.
The Banach space "B" has the Radon–Nikodym property if "B" has the Radon–Nikodym property with respect to every finite measure. It is known that the space formula_25 has the Radon–Nikodym property, but formula_26 and the spaces formula_27, formula_28, for formula_29 an open bounded subset of formula_30, and formula_31, for "K" an infinite compact space, do not. Spaces with Radon–Nikodym property include separable dual spaces (this is the Dunford–Pettis theorem) and reflexive spaces, which include, in particular, Hilbert spaces.