Measure theory

Positive and negative sets

In measure theory, given a measurable space and a signed measure on it, a set is called a positive set for if every -measurable subset of has nonnegative measure; that is, for every that satisfies holds. Similarly, a set is called a negative set for if for every subset satisfying holds. Intuitively, a measurable set is positive (resp. negative) for if is nonnegative (resp. nonpositive) everywhere on Of course, if is a nonnegative measure, every element of is a positive set for In the light of Radon–Nikodym theorem, if is a σ-finite positive measure such that a set is a positive set for if and only if the Radon–Nikodym derivative is nonnegative -almost everywhere on Similarly, a negative set is a set where -almost everywhere. (Wikipedia).

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This tutorial uses basic math and logic to demonstrate that a negative times a negative equals a positive. Join this channel to get access to perks: https://www.youtube.com/channel/UCn2SbZWi4yTkmPUj5wnbfoA/join :)

From playlist Basic Math

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From playlist Introduction to Integers

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Stanford EE104: Introduction to Machine Learning | 2020 | Lecture 12 - classifiers

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Hahn decomposition theorem | Signed measure | If and only if | Measure (mathematics) | Partition of a set | Null set | Radon–Nikodym theorem | Up to