condDistrib_const — Mathlib

English

Alternate expression for condDistrib with a constant function: condDistrib (fun _ => c) X μ equals the deterministic kernel at c.

Русский

Альтернативное выражение condDistrib для константной функции: condDistrib (функция, возвращающая c) X μ равнаDeterministic ядру в c.

LaTeX

$$$$ \\mathrm{condDistrib}(\\lambda x. c, X, μ) =_{a.e.} \\mathrm{deterministic}(\\lambda x. c). $$$$

Lean4

theorem condDistrib_const (X : α → β) (c : Ω) :
    condDistrib (fun _ ↦ c) X μ =ᵐ[μ.map X] Kernel.deterministic (fun _ ↦ c) (by fun_prop) :=
  by
  have : (fun _ : α ↦ c) = (fun _ : β ↦ c) ∘ X := rfl
  rw [this]
  filter_upwards [condDistrib_comp_self X (measurable_const (a := c))] with b hb
  rw [hb]

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