Theorems · Theorem · measure theory

MeasureTheory.setIntegral_ge_of_const_le

∀ {X : Type u_1} {E : Type u_3} {mX : MeasurableSpace X} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E]
  [inst_2 : PartialOrder E] [IsOrderedAddMonoid E] [IsOrderedModule ℝ E] {μ : MeasureTheory.Measure X} {f : X → E}
  {s : Set X} [ClosedIciTopology E] [CompleteSpace E] {c : E},
  MeasurableSet s →
    μ s ≠ ⊤ → (∀ x ∈ s, c ≤ f x) → MeasureTheory.IntegrableOn (fun x => f x) s μ → μ.real s • c ≤ ∫ (x : X) in s, f x ∂μ

Defined in

Mathlib.MeasureTheory.Integral.Bochner.Set

Cited by

1 results in Mathlib

Foundations

Depth 256 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpacePartialOrderIsOrderedAddMonoidIsOrderedModuleClosedIciTopologyCompleteSpace