integral_eq_zero_of_mul_left_eq_neg

English

If volume on α is sigma-finite and a fundamental domain exists, then the quotient measure on α/G is sigma-finite.

Русский

Если объём на α сигма-конечен и есть фундаментальная область, то мера на α/G сигма-конечна.

LaTeX

$$$[SigmaFinite(\\text{volume})] [HasFundamentalDomain G α] (μ) [QuotientMeasureEqMeasurePreimage volume μ] ⇒ SigmaFinite μ$$$

Lean4

/-- If some left-translate of a function negates it, then the integral of the function with respect
to a left-invariant measure is 0. -/
@[to_additive /-- If some left-translate of a function negates it, then the integral of the function with
          respect to a left-invariant measure is 0. -/
    ]
theorem integral_eq_zero_of_mul_left_eq_neg [IsMulLeftInvariant μ] (hf' : ∀ x, f (g * x) = -f x) : ∫ x, f x ∂μ = 0 := by
  simp_rw [← self_eq_neg ℝ E, ← integral_neg, ← hf', integral_mul_left_eq_self]

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