Sub-probability measure
In the mathematical theory of probability and measure, a sub-probability measure is a measure that is closely related to probability measures. 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 be a measure on the measurable space .
Then is called a sub-probability measure if .[1][2]
Properties
In measure theory, the following implications hold between measures:
So every probability measure is a sub-probability measure, but the converse is not true. Also every sub-probability measure is a finite measure and a σ-finite measure, but the converse is again not true.
See also
References
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Measure theory
- Absolute continuity of measures
- Lebesgue integration
- Lp spaces
- Measure
- Measure space
- Probability space
- Measurable space/function
- Almost everywhere
- Atom
- Baire set
- Borel set
- equivalence relation
- Borel space
- Carathéodory's criterion
- Cylindrical σ-algebra
- 𝜆-system
- Essential range
- Locally measurable
- π-system
- σ-algebra
- Non-measurable set
- Null set
- Support
- Transverse measure
- Universally measurable
- Atomic
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- Content
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- Finite
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- Locally finite
- Maximising
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- Pre-measure
- (Sub-) Probability
- Projection-valued
- Radon
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- Set function
- σ-finite
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- Vector
- Carathéodory's extension theorem
- Convergence theorems
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- Radon–Nikodym
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