Countable Grothendieck--Bessel Gram theorem

Adapted for Lean Pool from the Apache-2.0 upstream formalization, with module-system, Mathlib API, and compilation changes.

theorem GrothendieckBessel.main_proof (H : Type u_1) [NormedAddCommGroup H] [InnerProductSpace ℂ H] (x : ℕ → H) (hx : BesselSequenceBoundOne x) : ∃ (f : ℕ → ℝ → ℂ), (∀ (j : ℕ), MeasureTheory.AEStronglyMeasurable (f j) μ) ∧ (∀ (j : ℕ), ∀ t ∈ unitInterval, ‖f j t‖ ≤ 1) ∧ ∀ (j k : ℕ), inner ℂ (x k) (x j) = ∫ (t : ℝ) in unitInterval, f j t * star (f k t)

Proof implementation for the single TeX-shaped public theorem.