enorm_indicatorConstLp_le — Mathlib

English

The Lp edistance between two indicatorConstLp functions equals the ENNReal norm of the indicator on the symmetric difference.

Русский

Eдистант между двумя indicatorConstLp равен ENNReal-модулю индикатора на симметрической разности множеств.

LaTeX

$$$edist\big(indicatorConstLp(p, hs, hμs, c),\ indicatorConstLp(p, ht, hμt, c)\big) = \|indicatorConstLp(p, hs.symmDiff(ht), measure_symmDiff_ne_top(hμs, hμt), c)\|_e.$$$

Lean4

theorem enorm_indicatorConstLp_le : ‖indicatorConstLp p hs hμs c‖ₑ ≤ ‖c‖ₑ * μ s ^ (1 / p.toReal) := by
  simpa [ENNReal.coe_rpow_of_nonneg, ENNReal.coe_toNNReal hμs, Lp.enorm_def, ← enorm_eq_nnnorm] using
    ENNReal.coe_le_coe.2 <| nnnorm_indicatorConstLp_le (c := c) (hμs := hμs)

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