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Measure Theory (Definition)

The branch of analysis concerned with assigning a notion of size (measure) to subsets of a set.
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The statement of the theorem

Let X\text{X} be a non-empty set. A σ\sigma-algebra A\text{A} on X\text{X} is a collection of subsets of X\text{X} satisfying: (i) X∈A\text{X} \in \text{A}; (ii) X∖E∈A\text{X} \setminus E \in \text{A} for all E∈AE \in \text{A}; and (iii) \text{\bigcup}_{i=1}^{\infty} E_i \in \text{A} for any countable sequence (Ei)i=1∞(E_i)_{i=1}^{\infty} with Ei∈AE_i \in \text{A}. The pair (X,A)(\text{X}, \text{A}) is called a measurable space.\n\nGiven a measurable space (X,A)(\text{X}, \text{A}), a measure μ\mu is a function μ:A→[0,∞]\mu: \text{A} \to [0, \infty] such that: (i) μ(∅)=0\mu(\emptyset) = 0; (ii) μ\mu is countably additive: for any sequence of pairwise disjoint sets (Ei)i=1∞(E_i)_{i=1}^{\infty} in A\text{A}, we have μ(⋃i=1∞Ei)=∑i=1∞μ(Ei)\mu(\bigcup_{i=1}^{\infty} E_i) = \sum_{i=1}^{\infty} \mu(E_i).\n\nIf X=[a,b]⊂R\text{X} = [a, b] \subset \mathbb{R} and A=B[a,b]\text{A} = \mathcal{B}_{[a, b]} (the Borel σ\sigma-algebra restricted to [a,b][a, b]), the Lebesgue measure λ\lambda is the unique measure (up to scaling) satisfying λ([x,y])=y−x\lambda([x, y]) = y - x for all x,y∈[a,b]x, y \in [a, b]. The resulting structure (X,A,μ)(\text{X}, \text{A}, \mu) is the foundation of Measure Theory.