English
The Haar measure on K0 assigns mass 1 to K0: haarMeasure K0 K0 = 1.
Русский
Мера Хаара на K0 придает массе K0 равной 1: haarMeasure K0 K0 = 1.
LaTeX
$$haarMeasure_K0(K0) = 1$$
Lean4
@[to_additive]
theorem haarMeasure_self {K₀ : PositiveCompacts G} : haarMeasure K₀ K₀ = 1 :=
by
haveI : LocallyCompactSpace G := K₀.locallyCompactSpace_of_group
simp only [haarMeasure, coe_smul, Pi.smul_apply, smul_eq_mul]
rw [← K₀.isCompact.measure_closure, Content.measure_apply _ isClosed_closure.measurableSet, ENNReal.inv_mul_cancel]
· exact (haarContent_outerMeasure_closure_pos K₀).ne'
· exact (Content.outerMeasure_lt_top_of_isCompact _ K₀.isCompact.closure).ne