Mathlib theorem · Existing formal mathematics
ProofAtlas

Existing in Mathlib · Probability and analysis

Chebyshev Inequality

Fix a positive distance c from the integral reference μ[X]. The larger that distance is relative to the recorded variance expression, the smaller the set on which X can deviate by at least c. Mathlib states the estimate for any finite measure and a real-valued L² function; only after specializing μ to a probability measure is μ[X] the usual expectation and the measured set a tail probability.

Exact theorem

Exact Mathlib statement

theorem ProbabilityTheory.meas_ge_le_variance_div_sq {Ω : Type*} {mΩ : MeasurableSpace Ω} {μ : Measure Ω} [IsFiniteMeasure μ] {X : Ω → ℝ} (hX : MemLp X 2 μ) {c : ℝ} (hc : 0 < c) : μ {ω | c ≤ |X ω - μ[X]|} ≤ ENNReal.ofReal (variance X μ / c ^ 2)

The theorem at a glance

Chebyshev inequality at a glance

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

For a real L² function under a finite measure, deviations of size at least c have measure at most variance divided by c². Explanatory diagram.
Detailed visual description

The poster introduces the finite measure, real L² function, integral reference μ[X], variance expression, and positive threshold. A central band around μ[X] leaves two outer deviation regions, which are collected into the event measured on the left side of the inequality. A short proof route passes from L² regularity to extended variance and then to the real-variance form.

Statement structure

Statement and scope

Statement map for Chebyshev InequalityA real L² function cannot spend too much finite measure far from the integral reference μ[X]: the measure of deviations of size at least c is bounded by the recorded variance expression divided by c². Claim boundary: This target indexes Mathlib's finite-measure, real-valued L² form of Chebyshev's inequality. For c > 0 it bounds the μ-measure of {ω | c ≤ |X ω - μ[X]|} by ENNReal.ofReal (variance X μ / c²). It does not add a probability-measure hypothesis, an independence hypothesis, a distributional model, a sharpness assertion, or an asymptotic concentration rate. The pinned upstream declaration is ProbabilityTheory.meas_ge_le_variance_div_sq. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingA real L² functioncannot spend too muchfinite measure far fromthe integral referenceμ[X]: the measure ofdeviations of size atleast c is bounded bythe recorded varianceexpression divided byc².Claim boundaryThis target indexesMathlib'sfinite-measure,real-valued L² form ofChebyshev's inequality.For c > 0 it bounds theμ-measure of {ω | c ≤ |Xω - μ[X]|} byENNReal.ofReal (varianceX μ / c²). It does notadd aprobability-measurehypothesis, anindependence hypothesis,a distributional model,a sharpness assertion,or an asymptoticconcentration rate.Pinned declarationmathlib ·ProbabilityTheory.meas_ge_le_variance_div_sqStatement map for Chebyshev InequalityA real L² function cannot spend too much finite measure far from the integral reference μ[X]: the measure of deviations of size at least c is bounded by the recorded variance expression divided by c². Claim boundary: This target indexes Mathlib's finite-measure, real-valued L² form of Chebyshev's inequality. For c > 0 it bounds the μ-measure of {ω | c ≤ |X ω - μ[X]|} by ENNReal.ofReal (variance X μ / c²). It does not add a probability-measure hypothesis, an independence hypothesis, a distributional model, a sharpness assertion, or an asymptotic concentration rate. The pinned upstream declaration is ProbabilityTheory.meas_ge_le_variance_div_sq. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingA real L² functioncannot spend too muchfinite measure far fromthe integral referenceμ[X]: the measure ofdeviations of size atleast c is bounded bythe recorded varianceexpression divided byc².Claim boundaryThis target indexesMathlib'sfinite-measure,real-valued L² form ofChebyshev's inequality.For c > 0 it bounds theμ-measure of {ω | c ≤ |Xω - μ[X]|} byENNReal.ofReal (varianceX μ / c²). It does notadd aprobability-measurehypothesis, anindependence hypothesis,a distributional model,a sharpness assertion,or an asymptoticconcentration rate.Pinned declarationmathlib ·ProbabilityTheory.meas_ge_le_variance_div_sq

Read the exact Mathlib declaration

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

For a real L² function under any finite measure, a positive threshold bounds the measure of two-sided deviations from the integral reference μ[X] by the recorded variance expression divided by c². Explanatory scientific diagram.
Detailed visual description

The exact theorem centers deviations on the unnormalized integral reference μ[X], places equal positive thresholds on both sides, and measures the two outer deviation regions together. A second-moment budget passes through a square-threshold bottleneck before controlling those regions. The statement assumes only a finite measure, so the central reference is not a normalized mean and the outer measure is not a probability unless μ is specialized to a probability measure. It does not assert equality, sharpness, independence, a distributional model, or asymptotic concentration.

Why it matters

A mathematical landmark

Chebyshev's inequality is one of the basic universal concentration bounds in probability and statistics. It converts a second-moment quantity into a tail bound without independence, distributional shape, or higher-moment assumptions, and it is a standard bridge from variance estimates to convergence in probability.

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 target indexes Mathlib's finite-measure, real-valued L² form of Chebyshev's inequality. For c > 0 it bounds the μ-measure of {ω | c ≤ |X ω - μ[X]|} by ENNReal.ofReal (variance X μ / c²). It does not add a probability-measure hypothesis, an independence hypothesis, a distributional model, a sharpness assertion, or an asymptotic concentration rate.

Source and local evidence

Where the theorem comes from

Existing declaration
ProbabilityTheory.meas_ge_le_variance_div_sq in mathlib
Relationship
The checked artifact is the upstream declaration itself
Local Lean evidence
artifact.library.mathlib.chebyshev-inequality.v001
Source
Open the pinned upstream reference