Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Chebyshev-Markov inequality for the integral

Statement

Let f:X→[0,+∞] be measurable and let t>0. Then μ({f≥t})≤1t∫f dμ.

Facts & Assumptions

Given: A nonnegative measurable function f and a real number t>0.

[L1]

The nonnegative integral is monotone and homogeneous (Monotonicity and nonnegative homogeneity of the nonnegative integral).

Proof

technique · direct
1.1

If E:={f≥t}, then tχE≤f. Applying [L1] gives[L1, given] t μ(E)=∫tχE dμ≤∫f dμ.

2.1

Dividing by the positive number t yields[step 1.1, algebra] ∎ μ({f≥t})≤t−1∫f dμ.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources