Theorems · Definition · real analysis

BoxIntegral.Integrable.toBoxAdditive

{ι : Type u} →
  {E : Type v} →
    {F : Type w} →
      [inst : NormedAddCommGroup E] →
        [inst_1 : NormedSpace ℝ E] →
          [inst_2 : NormedAddCommGroup F] →
            [inst_3 : NormedSpace ℝ F] →
              {I : BoxIntegral.Box ι} →
                [inst_4 : Fintype ι] →
                  {l : BoxIntegral.IntegrationParams} →
                    {f : (ι → ℝ) → E} →
                      {vol : BoxIntegral.BoxAdditiveMap ι (E →L[ℝ] F) ⊤} →
                        [CompleteSpace F] → BoxIntegral.Integrable I l f vol → BoxIntegral.BoxAdditiveMap ι F ↑I

If f is integrable on I, then fun J ↦ integral J l f vol is box-additive on subboxes of I: if π₁, π₂ are two prepartitions of I covering the same part of I, the sum of integrals of f over the boxes of π₁ is equal to the sum of integrals of f over the boxes of π₂. See also BoxIntegral.Integrable.sum_integral_congr for an unbundled version.

Defined in

Mathlib.Analysis.BoxIntegral.Basic

Cited by

2 results in Mathlib

Foundations

Depth 183 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceFintypeCompleteSpace