feat(Topology/Algebra/InfiniteSum/Real): Add partition lemma (#12446)

We also  `summable_partition` (and the required  `Set.sigmaEquiv`), needed for #10377.  



Co-authored-by: Chris Birkbeck <c.birkbeck@uea.ac.uk>

Mathlib/Data/Set/Lattice.lean

@@ -2197,6 +2197,15 @@ theorem sigmaToiUnion_bijective (h : Pairwise fun i j => Disjoint (t i) (t j)) :
   ⟨sigmaToiUnion_injective t h, sigmaToiUnion_surjective t⟩
 #align set.sigma_to_Union_bijective Set.sigmaToiUnion_bijective
 
+/-- Equivalence from the disjoint union of a family of sets forming a partition of `β`, to `β`
+itself. -/
+noncomputable def sigmaEquiv (s : α → Set β) (hs : ∀ b, ∃! i, b ∈ s i) :
+    (Σ i, s i) ≃ β where
+  toFun | ⟨_, b⟩ => b
+  invFun b := ⟨(hs b).choose, b, (hs b).choose_spec.1⟩
+  left_inv | ⟨i, b, hb⟩ => Sigma.subtype_ext ((hs b).choose_spec.2 i hb).symm rfl
+  right_inv _ := rfl
+
 /-- Equivalence between a disjoint union and a dependent sum. -/
 noncomputable def unionEqSigmaOfDisjoint {t : α → Set β}
     (h : Pairwise fun i j => Disjoint (t i) (t j)) :

Mathlib/Topology/Algebra/InfiniteSum/Real.lean

@@ -81,6 +81,11 @@ theorem summable_sigma_of_nonneg {β : α → Type*} {f : (Σ x, β x) → ℝ}
   exact mod_cast NNReal.summable_sigma
 #align summable_sigma_of_nonneg summable_sigma_of_nonneg
 
+lemma summable_partition {α β : Type*} {f : β → ℝ} (hf : 0 ≤ f) {s : α  → Set β}
+    (hs : ∀ i, ∃! j, i ∈ s j) : Summable f ↔
+      (∀ j, Summable fun i : s j ↦ f i) ∧ Summable fun j ↦ ∑' i : s j, f i := by
+  simpa only [← (Set.sigmaEquiv s hs).summable_iff] using summable_sigma_of_nonneg (fun _ ↦ hf _)
+
 theorem summable_prod_of_nonneg {f : (α × β) → ℝ} (hf : 0 ≤ f) :
     Summable f ↔ (∀ x, Summable fun y ↦ f (x, y)) ∧ Summable fun x ↦ ∑' y, f (x, y) :=
   (Equiv.sigmaEquivProd _ _).summable_iff.symm.trans <| summable_sigma_of_nonneg fun _ ↦ hf _