# Strong integral

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.

#### References

[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. |

#### Comments

See also vector measure; strong topology.

#### References

[a1] | J. Diestel and J.J. Uhl, Jr., “Vector measures”, Math. Surveys, 15, Amer. Math. Soc. (1977). |

**How to Cite This Entry:**

Strong integral.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Strong_integral&oldid=38660