zero_mlconvolution — Mathlib

English

For measurable functions f, g, k on a group with a left-invariant measure, the convolution is associative: f ⋆ g ⋆ k = (f ⋆ g) ⋆ k.

Русский

Для измеримых функций f, g, k на группе с левонеприводимой мерой свёртка ассоциативна: f ⋆ g ⋆ k = (f ⋆ g) ⋆ k.

LaTeX

$$$f, g, k:\\, G \\to \\mathbb{R}_{\\ge 0}^{\\infty},\\ hf, hg, hk \\text{ измеримы} \\Rightarrow f \\star g \\star k = (f \\star g) \\star k$$$

Lean4

/-- Convolution of the zero function with a function returns the zero function. -/
@[to_additive (attr := simp) /-- Convolution of the zero function with a function returns the zero function. -/
    ]
theorem zero_mlconvolution (f : G → ℝ≥0∞) (μ : Measure G) : 0 ⋆ₘₗ[μ] f = 0 := by ext; simp [mlconvolution]

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