fix doc

CHANGELOG_UNRELEASED.md

@@ -53,8 +53,6 @@
   + lemmas `lteD2rE`, `leeD2rE`
   + lemmas `lte_dD2rE`, `lee_dD2rE`
 
-### Changed
-
 - in `classical_sets.v`:
   + lemmas `setC_I`, `bigcup_subset`
 

theories/charge.v

@@ -13,7 +13,11 @@ Require Import esum measure realfun lebesgue_measure lebesgue_integral.
 (* NB: See CONTRIBUTING.md for an introduction to HB concepts and commands.   *)
 (*                                                                            *)
 (* This file contains a formalization of charges (a.k.a. signed measures) and *)
-(* their theory (Hahn decomposition theorem, etc.).                           *)
+(* their theory (Hahn decomposition theorem, Radon-Nikodym derivative, etc.). *)
+(*                                                                            *)
+(* Reference:                                                                 *)
+(* - Y. Ishiguro, R. Affeldt. The Radon-Nikodym Theorem and the Lebesgue-     *)
+(*   Stieltjes measure in Coq. Computer Software 41(2) 2024                   *)
 (*                                                                            *)
 (* ## Structures for functions on classes of sets                             *)
 (* ```                                                                        *)
@@ -59,6 +63,10 @@ Require Import esum measure realfun lebesgue_measure lebesgue_integral.
 (*                              decomposition nuPN: hahn_decomposition nu P N *)
 (*     charge_variation nuPN == variation of the charge nu                    *)
 (*                           := jordan_pos nuPN \+ jordan_neg nuPN            *)
+(*              induced intf == charge induced by a function f : T -> \bar R  *)
+(*                              R has type realType; T is a measurableType.   *)
+(*                              intf is a proof that f is integrable over     *)
+(*                              [set: T]                                      *)
 (*              'd nu '/d mu == Radon-Nikodym derivative of nu w.r.t. mu      *)
 (*                              (the scope is charge_scope)                   *)
 (* ```                                                                        *)
@@ -1072,12 +1080,12 @@ move=> /choice[F /= H].
 have mF i : measurable (F i) by have [] := H i.
 have : mu (lim_sup_set F) = 0.
   apply: lim_sup_set_cvg0 => //.
-  have h : \sum_(0 <= n < k) (1 / 2 ^+ n.+1)%:E @[k --> \oo] --> (1 : \bar R).
+  have h : \sum_(0 <= n < k) (1 / 2 ^+ n.+1)%:E @[k --> \oo] --> (1%E : \bar R).
     apply/fine_cvgP; split.
       apply: nearW => /= n; rewrite sum_fin_num//.
       by apply/allP => /= r /mapP[/= k _] ->.
-    under eq_fun do rewrite sumEFin.
     have := @cvg_geometric_series_half R 1 0; rewrite {1}/series/= expr0 divr1.
+    under [in X in _ -> X]eq_fun do rewrite sumEFin.
     by under eq_fun do under eq_bigr do rewrite addn1 natrX.
   apply: (@le_lt_trans _ _ (\sum_(0 <= n <oo) (1 / (2 ^ n.+1))%:E)).
     apply: lee_lim.
@@ -1094,8 +1102,8 @@ suff : charge_variation nuPN (lim_sup_set F) >= e%:E by exact: lt_le_trans.
 have echarge n : e%:E <= charge_variation nuPN (\bigcup_(j >= n) F j).
   have [_ _ /le_trans] := H n; apply.
   rewrite le_measure// ?inE//; first exact: bigcup_measurable.
-  by apply: bigcup_sup => //=.
-have /(_ _ _)/cvg_lim <-// := @lim_sup_set_cvg _ _ _ (charge_variation nuPN) F.
+  by apply: bigcup_sup => /=.
+have /(_ _ _)/cvg_lim <-// := lim_sup_set_cvg (charge_variation nuPN) F.
   apply: lime_ge.
     apply: ereal_nonincreasing_is_cvgn => a b ab.
     rewrite le_measure ?inE//; [exact: bigcup_measurable|

theories/ftc.v

@@ -10,6 +10,12 @@ Require Import derive charge.
 (**md**************************************************************************)
 (* # Fundamental Theorem of Calculus for the Lebesgue Integral                *)
 (*                                                                            *)
+(* NB: See CONTRIBUTING.md for an introduction to HB concepts and commands.   *)
+(*                                                                            *)
+(* This file provides a proof of the first fundamental theorem of calculus    *)
+(* for the Lebesgue integral. We derive from this theorem a corollary to      *)
+(* compute the definite integral of continuous functions.                     *)
+(*                                                                            *)
 (* parameterized_integral mu a x f := \int[mu]_(t \in `[a, x] f t)            *)
 (*                                                                            *)
 (******************************************************************************)
@@ -341,7 +347,7 @@ Unshelve. all: by end_near. Qed.
 
 Lemma parameterized_integral_continuous a b (f : R -> R) : a < b ->
   mu.-integrable `[a, b] (EFin \o f) ->
-  {within `[a,b], continuous (fun x => int a x f)}.
+  {within `[a, b], continuous (fun x => int a x f)}.
 Proof.
 move=> ab intf; apply/(continuous_within_itvP _ ab); split; last first.
   split; last exact: parameterized_integral_cvg_at_left.

theories/lebesgue_stieltjes_measure.v

@@ -13,8 +13,10 @@ Require Import real_interval measure realfun.
 (* This file contains a formalization of the Lebesgue-Stieltjes measure using *)
 (* the Measure Extension theorem from measure.v.                              *)
 (*                                                                            *)
-(* Reference:                                                                 *)
+(* References:                                                                *)
 (* - Achim Klenke, Probability Theory 2nd edition, 2014                       *)
+(* - Y. Ishiguro, R. Affeldt. The Radon-Nikodym Theorem and the Lebesgue-     *)
+(*   Stieltjes measure in Coq. Computer Software 41(2) 2024                   *)
 (*                                                                            *)
 (* ```                                                                        *)
 (*           right_continuous f == the function f is right-continuous         *)

theories/measure.v

@@ -3701,7 +3701,7 @@ End measure_continuity.
 Definition lim_sup_set T (F : (set T)^nat) := \bigcap_n \bigcup_(j >= n) F j.
 
 Section borel_cantelli_realFieldType.
-Context d (T : measurableType d) {R : realFieldType}
+Context {d} {T : measurableType d} {R : realFieldType}
         (mu : {measure set T -> \bar R}).
 Implicit Types F : (set T)^nat.
 Local Open Scope ereal_scope.
@@ -3728,6 +3728,7 @@ move=> mF mFoo; apply: nonincreasing_cvg_mu => //.
 Qed.
 
 End borel_cantelli_realFieldType.
+Arguments lim_sup_set_cvg {d T R} mu F.
 
 Section borel_cantelli.
 Context d (T : measurableType d) {R : realType} (mu : {measure set T -> \bar R}).