View source for Strong integral

An integral of Lebesgue type either of a function with values in a linear topological space with respect to a scalar measure, or of a scalar function with respect to a measure with values in a vector space. 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|Bochner integral]] of a vector-valued function;

2) the [[Daniell integral|Daniell integral]], if the values of the integrand belong to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090570/s0905701.png" />-complete vector lattice;

3) the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090570/s0905702.png" />, giving the spectral decomposition of a self-adjoint operator acting on a Hilbert space (cf. [[Spectral decomposition of a linear operator|Spectral decomposition of a linear operator]]).

For the strong integral of scalar functions with respect to a vector measure, the values of the measure, in many cases, are assumed to belong to a semi-ordered vector space (cf. [[Semi-ordered space|Semi-ordered space]]).

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators. Spectral theory" , '''2''' , Interscience  (1963)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T.H. Hildebrandt,   "Integration in abstract spaces"  ''Bull. Amer. Math. Soc.'' , '''59'''  (1953)  pp. 111–139</TD></TR></table>



====Comments====
See also [[Vector measure|Vector measure]]; [[Strong topology|Strong topology]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Diestel,   J.J. Uhl jr.,   "Vector measures" , ''Math. Surveys'' , '''15''' , Amer. Math. Soc.  (1977)</TD></TR></table>