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
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 : ℝ => 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