charFun_conv — Mathlib

English

The Dirac measure χ_{dirac x} is given by χ_{dirac x}(t) = e^{i⟨x,t⟩}.

Русский

Характеристическая функция дираковской меры χ_{dirac x}(t) = e^{i⟨x,t⟩}.

LaTeX

$$$$\\chi_{\\mathrm{dirac}\\,x}(t) = e^{i \\langle x,t \\rangle}.$$$$

Lean4

/-- The characteristic function of a convolution of measures
is the product of the respective characteristic functions. -/
theorem charFun_conv [IsFiniteMeasure μ] [IsFiniteMeasure ν] (t : E) : charFun (μ ∗ ν) t = charFun μ t * charFun ν t :=
  by
  simp_rw [charFun_apply]
  rw [integral_conv]
  · simp [inner_add_left, add_mul, Complex.exp_add, integral_const_mul, integral_mul_const]
  · exact (integrable_const (1 : ℝ)).mono (by fun_prop) (by simp)

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