Integral
Linearity

The collection of Riemann-integrable functions on a closed interval forms a vector space under the operations of pointwise addition and multiplication by a scalar, and the operation of integration

is a linear functional on this vector space. Thus, the collection of integrable functions is closed under taking linear combinations, and the integral of a linear combination is the linear combination of the integrals:

Similarly, the set of real -valued Lebesgue-integrable functions on a given measure space E with measure μ is closed under taking linear combinations and hence form a vector space, and the Lebesgue integral

is a linear functional on this vector space, so that:

More generally, consider the vector space of all measurable functions on a measure space (E, μ), taking values in a locally compact complete topological vector space V over a locally compact topological field K, f : E → V. Then one may define an abstract integration map assigning to each function f an element of V or the symbol ∞,

that is compatible with linear combinations. In this situation, the linearity holds for the subspace of functions whose integral is an element of V (i.e. "finite"). The most important special cases arise when K is R, C, or a finite extension of the field Q of p-adic numbers, and V is a finite-dimensional vector space over K, and when K = C and V is a complex Hilbert space.

Linearity, together with some natural continuity properties and normalization for a certain class of "simple" functions, may be used to give an alternative definition of the integral. This is the approach of Daniell for the case of real-valued functions on a set X, generalized by Nicolas Bourbaki to functions with values in a locally compact topological vector space. See Hildebrandt 1953 for an axiomatic characterization of the integral.
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