Measure Theory

σ-Algebras, Integrals, and Measures.

Measure Theory & Lebesgue Integration Handbook

Mathematical Principles & Theorems

Measure theory formalizes generalized volume on \(\sigma\)-algebras. Lebesgue measure \(\lambda\) on \(\mathbb{R}\) extends interval length \(\lambda([a,b]) = b-a\). Unlike the Riemann integral which partitions the domain \([a,b]\), the Lebesgue integral \(\int_E f d\mu = \sup \sum y_i \mu(E_i)\) partitions the range (codomain), enabling integration of nowhere-continuous functions such as the Dirichlet indicator function \(\mathbf{1}_{\mathbb{Q}}\). Dominated Convergence Theorem guarantees limit-integral interchange.

Operating Instructions

Select a test function \(f(x)\): Continuous polynomial, discontinuous Step function, or Dirichlet indicator.
Adjust partition subdivision count \(N\) using the precise numeric value box.
Toggle between Riemann vertical domain slicing and Lebesgue horizontal range measure slicing.
Compare the convergence of Riemann Darboux sums versus Lebesgue integral measure sums.

Parameters

Set size n (X = {1,...,n})

σ-algebra axioms:
1. X ∈ Σ (whole space)
2. A ∈ Σ ⟹ Aᶜ ∈ Σ (closed under complement)
3. A₁,A₂,... ∈ Σ ⟹ ∪Aᵢ ∈ Σ (closed under countable union)

Smallest σ-algebras:
Trivial: {∅, X} | Every algebra: {∅, {a}, {b,...}, X, ...}
Borel σ-algebra ℬ(ℝ): generated by open intervals (a,b)
Contains all open, closed, Fσ, Gδ sets. Cannot be explicitly described.