tsum_fiberwise — Mathlib

English

Variant of Markov’s inequality: given nonnegative ENNReal a_i with tsum ≤ c, there exists a finite set whose cardinal is bounded by c/ε.

Русский

Вариант неравенства Маркова: при сумме сумм ≤ c существует конечное множество индексов, размер которого ≤ c/ε.

LaTeX

$$Same as above with coe adjustments; ∃hf : {i | ε ≤ a_i}.Finite, #hf ≤ c/ε$$

Lean4

theorem tsum_fiberwise (f : β → ℝ≥0∞) (g : β → γ) : ∑' x, ∑' b : g ⁻¹' { x }, f b = ∑' i, f i :=
  by
  apply HasSum.tsum_eq
  let equiv := Equiv.sigmaFiberEquiv g
  apply (equiv.hasSum_iff.mpr ENNReal.summable.hasSum).sigma
  exact fun _ ↦ ENNReal.summable.hasSum_iff.mpr rfl

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