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AlexKontorovich/PrimeNumberTheoremAnd
Source indexedtheorem · leanprover/lean4:v4.32.0

Li2Bounds.g_intervalIntegrable_full

PrimeNumberTheoremAnd.IEANTN.Li2Bounds · PrimeNumberTheoremAnd/IEANTN/Li2Bounds.lean:86 to 97

Source documentation

g is integrable on [0, 1]. Uses boundedness by 2 from Li2Bounds.

Exact Lean statement

theorem g_intervalIntegrable_full : IntervalIntegrable g volume 0 1

Complete declaration

Lean source

Canonical source
Full Lean sourceLean 4
theorem g_intervalIntegrable_full : IntervalIntegrable g volume 0 1 := by  apply (intervalIntegrable_iff_integrableOn_Ioo_of_le (by linarith)).2  apply Measure.integrableOn_of_bounded measure_Ioo_lt_top.ne  · exact Measurable.aestronglyMeasurable (by      unfold g      change Measurable fun t :  => 1 / log (1 + t) + 1 / log (1 - t)      fun_prop)  · filter_upwards [self_mem_ae_restrict (by measurability)] with t ht    have habs : |g t| = g t := abs_of_pos <|      LeanCert.CertifiedBounds.Li2.integrand_pos t ht.1 ht.2    simpa [Real.norm_eq_abs, habs] using!      LeanCert.CertifiedBounds.Li2.integrand_le_two t ht.1 ht.2