Bochner integral [SEP] named after
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 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
Pettis integral
In mathematics, the Pettis integral or Gelfand–Pettis integral, named after Israel M. Gelfand and Billy James Pettis, extends the definition of the Lebesgue integral to vector-valued functions on a measure space, by exploiting duality. The integral was introduced by Gelfand for the case when the measure space is an interval with Lebesgue measure. The integral is also called the weak integral in contrast to the Bochner integral, which is the strong integral.
Definition.
Let formula_1 where formula_2 is a measure space and