∀ {E : Type u_1} {mE : MeasurableSpace E} {S : Set (MeasureTheory.Measure E)} [inst : NormedAddCommGroup E]
  {𝕜 : Type u_2} [inst_1 : RCLike 𝕜] [inst_2 : InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E],
  (∀ (y : E), Filter.Tendsto (fun r => ⨆ μ ∈ S, μ {x | r < ‖inner 𝕜 y x‖}) Filter.atTop (nhds 0)) →
    MeasureTheory.IsTightMeasureSet S