setToSimpleFunc_mono_left' — Mathlib

English

If there is a structure SMul on the codomain, the smul commutes with setToSimpleFunc in the sense described by setToSimpleFunc_smul.

Русский

Если есть структура SMul на кодом-пространстве, умножение на скаляр commuting с setToSimpleFunc так, как описано в setToSimpleFunc_smul.

LaTeX

$$$ setToSimpleFunc_smul {E} \dots$$$

Lean4

theorem setToSimpleFunc_mono_left' (T T' : Set α → E →L[ℝ] G'')
    (hTT' : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, T s x ≤ T' s x) (f : α →ₛ E) (hf : Integrable f μ) :
    setToSimpleFunc T f ≤ setToSimpleFunc T' f :=
  by
  refine sum_le_sum fun i _ => ?_
  by_cases h0 : i = 0
  · simp [h0]
  · exact hTT' _ (measurableSet_fiber _ _) (measure_preimage_lt_top_of_integrable _ hf h0) i

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