integral_pos_of_hasCompactSupport_nonneg_nonzero

English

For a subset s, restricting f in Lp to s does not increase its norm.

Русский

Для множества s ограничение f в Lp на s не увеличивает норму.

LaTeX

$$$$\\|((Lp.memLp f).restrict s).toLp f\\| ≤ \\|f\\|.$$$$

Lean4

theorem integral_pos_of_hasCompactSupport_nonneg_nonzero [IsFiniteMeasureOnCompacts μ] {f : X → ℝ} {x : X}
    (f_cont : Continuous f) (f_comp : HasCompactSupport f) (f_nonneg : 0 ≤ f) (f_x : f x ≠ 0) : 0 < ∫ x, f x ∂μ :=
  integral_pos_of_integrable_nonneg_nonzero f_cont (f_cont.integrable_of_hasCompactSupport f_comp) f_nonneg f_x

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