Existing in Mathlib · Probability and analysis

Strong Law of Large Numbers

For almost every outcome, the running average of pairwise independent, identically distributed integrable real random variables settles to their common expected value.

pairwise independence
identical distribution
integrability
empirical averages
almost-sure convergence

Exact theorem

Exact Mathlib statement

theorem ProbabilityTheory.strong_law_ae_real {Ω : Type*} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} (X : ℕ → Ω → ℝ) (hint : MeasureTheory.Integrable (X 0) μ) (hindep : Pairwise ((· ⟂ᵢ[μ] ·) on X)) (hident : ∀ i, ProbabilityTheory.IdentDistrib (X i) (X 0) μ μ) : ∀ᵐ ω ∂μ, Filter.Tendsto (fun n : ℕ => (∑ i ∈ Finset.range n, X i ω) / n) Filter.atTop (𝓝 μ[X 0])

The theorem at a glance

Strong Law of Large Numbers at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

An ivory probability poster shows pairwise-independent observation streams feeding running averages that settle toward a common horizontal expectation line, with an almost-everywhere scope marker. — Pairwise-independent identically distributed integrable real variables have empirical averages converging almost everywhere to their common expectation.

Statement structure

Statement and scope

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

Many pairwise-linked random observations feed a running-average ribbon whose fluctuations narrow around one gold expectation level while a negligible set fades away. — Along almost every outcome, the empirical-average path settles onto the common expectation.

Why it matters

A mathematical landmark

The strong law is probability's fundamental pathwise stabilization theorem. It connects repeated observation with long-run average behavior and complements the collection's distributional Central Limit Theorem with a genuinely almost-sure conclusion.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This page indexes Mathlib's real-valued almost-sure strong law for an integrable, identically distributed sequence with pairwise independence. It concludes almost-everywhere convergence of empirical averages to the integral of X 0. It does not state a convergence rate, finite-sample concentration bound, Lp convergence, convergence in distribution only, or the more general Banach-valued theorem.

The selected declaration is real-valued; Mathlib's more general Banach-valued strong law is a separate declaration.
It provides no convergence rate, concentration inequality, or finite-sample threshold.
ProofAtlas indexes an existing Mathlib theorem; generated explanations and visuals are not proof evidence.

Source and local evidence

Where the theorem comes from

Existing declaration: ProbabilityTheory.strong_law_ae_real in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.strong-law-large-numbers.v001
Source: Open the pinned upstream reference