a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume

Measure - Properties

Let:

A set function, 𝜇 : 𝛴 → extended real number line, is called a measure if the following conditions hold:

𝜇(∅) = 0
non-negativity - for all 𝐸∊𝛴, 𝜇(𝐸) ≥ 0
countable additivity (or 𝜎-additivity) - for all countable collections {𝐸_𝑘}_{1≤𝑘≤∞} of pairwise disjoint sets in 𝛴:
- 𝜇(⋃_{1≤𝑘≤∞}𝐸_𝑘) = 𝛴_{1≤𝑘≤∞}𝜇(𝐸_𝑘)

Measure - Types

Probability Measure - has an additional property that it must assign the value 1 to the entire set 𝑋 (i.e. 𝜇(𝑋) = 1)
- Martingale / Risk-Neutral measure - is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure
Borel measure – measure defined on all open sets of a topological space
Fuzzy measure – theory of generalized measures in which the additive property is replaced by the weaker property of monotonicity
Haar measure – left-invariant (or right-invariant) measure on locally compact topological group
Lebesgue measure – the concept of area in any dimension