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| − | A concept in measure theory, determined by a certain topology in a space of measures that are defined on a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c0261401.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c0261402.png" /> of subsets of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c0261403.png" /> or, more generally, in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c0261404.png" /> of charges, i.e. countably-additive real or complex functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c0261405.png" />, defined on sets from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c0261406.png" />. The following are the most commonly used topologies in the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c0261407.png" /> consisting of bounded charges, i.e. charges for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c0261408.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c0261409.png" />.
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| − | 1) In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614010.png" /> the norm
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614011.png" /></td> </tr></table>
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| − | called the variation of the charge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614012.png" />, is introduced. The convergence of a sequence of charges <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614014.png" />, to a charge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614015.png" /> in this norm is called convergence in variation.
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| − | 2) In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614016.png" /> the ordinary weak topology is examined: Convergence of a sequence of charges <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614018.png" />, in this topology (weak convergence) means that for any continuous linear function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614019.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614022.png" />. This convergence is equivalent to the fact that the sequence of charges is bounded, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614023.png" />, and that for any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614024.png" /> the sequence of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614026.png" />. Weak convergence of a sequence of charges <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614028.png" /> implies convergence of the integrals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614030.png" />, for any bounded function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614031.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614032.png" /> that is measurable with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614033.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614034.png" />.
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| − | 3) When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614035.png" /> is a topological space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614036.png" /> is its Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614037.png" />-algebra, a topology is examined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614038.png" /> which is also called the weak topology (or sometimes the narrow topology). It is defined as the weakest of the topologies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614039.png" /> relative to which all functionals of the form
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614040.png" /></td> </tr></table>
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| − | are continuous, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614041.png" /> is an arbitrary bounded continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614042.png" />. This topology is weaker than the previous one, and convergence of a sequence of charges <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614044.png" />, relative to it (weak or narrow convergence) is equivalent to the convergence of the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614046.png" />, for any Borel set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614047.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614048.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614049.png" /> and the operation of closure of a set is denoted by the bar.
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| − | 4) When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614050.png" /> is a locally compact topological space (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614051.png" /> is a Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614052.png" />-algebra) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614053.png" /> the so-called wide topology is examined: the convergence of a sequence of charges <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614055.png" /> (wide convergence), means convergence of the functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614057.png" />, for any continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614058.png" /> with compact support. This topology is weaker than the weak topology in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614059.png" />. An analogous topology is defined naturally in the wider space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614060.png" /> of locally bounded charges <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614061.png" />, i.e. charges such that for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614062.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614063.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026140/c02614066.png" />.
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| − | ====References====
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| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Dunford, J.T. Schwartz, "Linear operators. General theory" , '''1''' , Interscience (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P. Billingsley, "Convergence of probability measures" , Wiley (1968)</TD></TR></table>
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