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In the mathematical theory of
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
and measure, a sub-probability measure is a measure that is closely related to
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more g ...
s. While probability measures always assign the value 1 to the underlying set, sub-probability measures assign a value lesser than or equal to 1 to the underlying set.


Definition

Let \mu be a measure on the
measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a σ-algebra \mathcal A on X. Then ...
(X, \mathcal A) . Then \mu is called a sub-probability measure if \mu(X) \leq 1 .


Properties

In measure theory, the following implications hold between measures: \text \implies \text \implies \text \implies \sigma\text So every probability measure is a sub-probability measure, but the converse is not true. Also every sub-probability measure is a
finite measure In measure theory, a branch of mathematics, a finite measure or totally finite measure is a special measure that always takes on finite values. Among finite measures are probability measures. The finite measures are often easier to handle than ...
and a σ-finite measure, but the converse is again not true.


See also

* Helly's selection theorem *
Helly–Bray theorem In probability theory, the Helly–Bray theorem relates the weak convergence of cumulative distribution functions to the convergence of expectations of certain measurable functions. It is named after Eduard Helly and Hubert Evelyn Bray. Let '' ...


References

{{Measure theory Probability theory Measures (measure theory)