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A measure concentrated on a set which is at most countable. More generally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033090/d0330901.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033090/d0330902.png" /> be measures (usually with alternating signs) defined on a semi-ring of sets (with its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033090/d0330903.png" />-ring of measurable sets). The measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033090/d0330904.png" /> is said to be a discrete measure with respect to the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033090/d0330905.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033090/d0330906.png" /> is concentrated on a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033090/d0330907.png" />-measure zero which is at most countable and any one-point subset of which is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033090/d0330908.png" />-measurable. For instance, the discrete Lebesgue–Stieltjes measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033090/d0330909.png" /> of linear sets is equal on half-intervals to the increment of some jump function, which is of bounded variation if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033090/d03309010.png" /> is bounded, and which is non-decreasing if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033090/d03309011.png" /> is non-negative.
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A measure concentrated on a set which is at most countable. More generally, let $\lambda$ and $\mu$ be measures (usually with alternating signs) defined on a semi-ring of sets (with its $\sigma$-ring of measurable sets). The measure $\lambda$ is said to be a discrete measure with respect to the measure $\mu$ if $\lambda$ is concentrated on a set of $\mu$-measure zero which is at most countable and any one-point subset of which is $\lambda$-measurable. For instance, the discrete Lebesgue–Stieltjes measure $\lambda$ of linear sets is equal on half-intervals to the increment of some jump function, which is of bounded variation if $\lambda$ is bounded, and which is non-decreasing if $\lambda$ is non-negative.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR></table>

Latest revision as of 18:23, 16 April 2014

A measure concentrated on a set which is at most countable. More generally, let $\lambda$ and $\mu$ be measures (usually with alternating signs) defined on a semi-ring of sets (with its $\sigma$-ring of measurable sets). The measure $\lambda$ is said to be a discrete measure with respect to the measure $\mu$ if $\lambda$ is concentrated on a set of $\mu$-measure zero which is at most countable and any one-point subset of which is $\lambda$-measurable. For instance, the discrete Lebesgue–Stieltjes measure $\lambda$ of linear sets is equal on half-intervals to the increment of some jump function, which is of bounded variation if $\lambda$ is bounded, and which is non-decreasing if $\lambda$ is non-negative.

References

[1] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
How to Cite This Entry:
Discrete measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Discrete_measure&oldid=31791
This article was adapted from an original article by A.P. Terekhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article