exists_le_average — Mathlib

English

Let μ ≠ 0 and f integrable; then there exists x with f x ≤ ⨍ a, f a ∂μ.

Русский

Пусть μ ≠ 0; существует x, где f(x) ≤ ⨍ a, f(a) ∂μ.

LaTeX

$$$$ \exists x, f(x) \leq \laverage f $$$$

Lean4

/-- **First moment method**. The minimum of an integrable function is smaller than its mean. -/
theorem exists_le_average (hμ : μ ≠ 0) (hf : Integrable f μ) : ∃ x, f x ≤ ⨍ a, f a ∂μ :=
  let ⟨x, hx⟩ := nonempty_of_measure_ne_zero (measure_le_average_pos hμ hf).ne'
  ⟨x, hx⟩

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