setToFun_top_smul_measure — Mathlib

English

Given c ≠ ∞ and hT, hT_smul, congruence holds: setToFun μ T hT f = setToFun (c·μ) T hT_smul f.

Русский

При неравенстве c ≠ ∞ сохраняется эквивалентность: setToFun от μ и c·μ совпадают.

LaTeX

$$$\\mathrm{setToFun}\\;\\mu\\;T\\;hT\\;f = \\mathrm{setToFun}\\; (c\\cdot\\mu)\\; T\\; hT_{\\text{smul}}\\; f$$$

Lean4

theorem setToFun_top_smul_measure (hT : DominatedFinMeasAdditive (∞ • μ) T C) (f : α → E) :
    setToFun (∞ • μ) T hT f = 0 :=
  by
  refine setToFun_measure_zero' hT fun s _ hμs => ?_
  rw [lt_top_iff_ne_top] at hμs
  simp only [true_and, Measure.smul_apply, ENNReal.mul_eq_top, top_ne_zero, Ne, not_false_iff, not_or,
    Classical.not_not, smul_eq_mul] at hμs
  simp only [hμs.right, Measure.smul_apply, mul_zero, smul_eq_mul]

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