# Discrete measure

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