Strong integral

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An integral of Lebesgue type, either of a function with values in a linear topological space with respect to a scalar-valued measure, or of a scalar-valued function with respect to a vector-valued measure. Here, the limit processes by which the integral is defined are taken in the sense of the strong topology. Examples of strong integrals are:

1) The Bochner integral of a vector-valued function.

2) The Daniell integral, if the values of the integrand belong to a $ \sigma $-complete vector lattice.

3) The integral $ \displaystyle \int_{- \infty}^{\infty} \lambda ~ \mathrm{d}{F_{\lambda}} $, giving the spectral decomposition of a self-adjoint operator acting on a Hilbert space.

For the strong integral of scalar-valued functions with respect to a vector-valued measure, the values of the measure, in many cases, are assumed to belong to a semi-ordered vector space.


[1a] N. Dunford and J.T. Schwartz, “Linear operators. General theory”, 1, Interscience (1958).
[1b] N. Dunford and J.T. Schwartz, “Linear operators. Spectral theory”, 2, Interscience (1963).
[2] T.H. Hildebrandt, “Integration in abstract spaces”, Bull. Amer. Math. Soc., 59 (1953), pp. 111–139.


See also vector measure; strong topology.


[a1] J. Diestel and J.J. Uhl, Jr., “Vector measures”, Math. Surveys, 15, Amer. Math. Soc. (1977).
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Strong integral. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article