ext_of_generate_finite — Mathlib

English

If μ is finite on compacts, then the NNReal-scalar multiple of μ is finite on compacts.

Русский

Если μ конечна на компактных, то скалярное умножение на NNReal сохраняет конечность на компактных.

LaTeX

$$IsFiniteMeasureOnCompacts (instHSMul.hSMul c μ)$$

Lean4

/-- Two finite measures are equal if they are equal on the π-system generating the σ-algebra
  (and `univ`). -/
theorem ext_of_generate_finite (C : Set (Set α)) (hA : m0 = generateFrom C) (hC : IsPiSystem C) [IsFiniteMeasure μ]
    (hμν : ∀ s ∈ C, μ s = ν s) (h_univ : μ univ = ν univ) : μ = ν :=
  Measure.ext fun _s hs => ext_on_measurableSpace_of_generate_finite m0 C hμν le_rfl hA hC h_univ hs

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