Difference between revisions of "Discrete measure"
From Encyclopedia of Mathematics
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| − | A measure concentrated on a set which is at most countable. More generally, let | + | {{TEX|done}} |
| + | 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=12919
Discrete measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Discrete_measure&oldid=12919
This article was adapted from an original article by A.P. Terekhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article