# Theory Essential_Supremum

(*  Author:  Sébastien Gouëzel   sebastien.gouezel@univ-rennes1.fr
Author:  Johannes Hölzl (TUM) -- ported to Limsup
*)

theory Essential_Supremum
imports
begin

lemma ae_filter_eq_bot_iff:

section ‹The essential supremum›

text ‹In this paragraph, we define the essential supremum and give its basic properties. The
essential supremum of a function is its maximum value if one is allowed to throw away a set
of measure $0$. It is convenient to define it to be infinity for non-measurable functions, as
it allows for neater statements in general. This is a prerequisiste to define the space $L^\infty$.›

definition esssup::"'a measure  ('a  'b::{second_countable_topology, dense_linorder, linorder_topology, complete_linorder})  'b"
where "esssup M f = (if f  borel_measurable M then Limsup (ae_filter M) f else top)"

lemma esssup_non_measurable: "f  M M borel  esssup M f = top"

lemma esssup_eq_AE:
assumes f: "f  M M borel" shows "esssup M f = Inf {z. AE x in M. f x  z}"
unfolding esssup_def if_P[OF f] Limsup_def
proof (intro antisym INF_greatest Inf_greatest; clarsimp)
fix y assume "AE x in M. f x  y"
then have "(λx. f x  y)  {P. AE x in M. P x}"
by simp
then show "(INF P{P. AE x in M. P x}. SUP xCollect P. f x)  y"
by (rule INF_lower2) (auto intro: SUP_least)
next
fix P assume P: "AE x in M. P x"
show "Inf {z. AE x in M. f x  z}  (SUP xCollect P. f x)"
proof (rule Inf_lower; clarsimp)
show "AE x in M. f x  (SUP xCollect P. f x)"
using P by (auto elim: eventually_mono simp: SUP_upper)
qed
qed

lemma esssup_eq: "f  M M borel  esssup M f = Inf {z. emeasure M {x  space M. f x > z} = 0}"
by (auto simp add: esssup_eq_AE not_less[symmetric] AE_iff_measurable[OF _ refl] intro!: arg_cong[where f=Inf])

lemma esssup_zero_measure:
"emeasure M {x  space M. f x > esssup M f} = 0"
proof (cases "esssup M f = top")
case True
then show ?thesis by auto
next
case False
then have f[measurable]: "f  M M borel" unfolding esssup_def by meson
have "esssup M f < top" using False by (auto simp: less_top)
have *: "{x  space M. f x > z}  null_sets M" if "z > esssup M f" for z
proof -
have "w. w < z  emeasure M {x  space M. f x > w} = 0"
using z > esssup M f f by (auto simp: esssup_eq Inf_less_iff)
then obtain w where "w < z" "emeasure M {x  space M. f x > w} = 0" by auto
then have a: "{x  space M. f x > w}  null_sets M" by auto
have b: "{x  space M. f x > z}  {x  space M. f x > w}" using w < z by auto
show ?thesis using null_sets_subset[OF a _ b] by simp
qed
obtain u::"nat  'b" where u: "n. u n > esssup M f" "u  esssup M f"
using approx_from_above_dense_linorder[OF esssup M f < top] by auto
have "{x  space M. f x > esssup M f} = (n. {x  space M. f x > u n})"
using u apply auto
apply (metis (mono_tags, lifting) order_tendsto_iff eventually_mono LIMSEQ_unique)
using less_imp_le less_le_trans by blast
also have "...  null_sets M"
using *[OF u(1)] by auto
finally show ?thesis by auto
qed

lemma esssup_AE: "AE x in M. f x  esssup M f"
proof (cases "f  M M borel")
case True then show ?thesis
by (intro AE_I[OF _ esssup_zero_measure[of _ f]]) auto

lemma esssup_pos_measure:
"f  borel_measurable M  z < esssup M f  emeasure M {x  space M. f x > z} > 0"
using Inf_less_iff mem_Collect_eq not_gr_zero by (force simp: esssup_eq)

lemma esssup_I [intro]: "f  borel_measurable M  AE x in M. f x  c  esssup M f  c"
unfolding esssup_def by (simp add: Limsup_bounded)

lemma esssup_AE_mono: "f  borel_measurable M  AE x in M. f x  g x  esssup M f  esssup M g"
by (auto simp: esssup_def Limsup_mono)

lemma esssup_mono: "f  borel_measurable M  (x. f x  g x)  esssup M f  esssup M g"
by (rule esssup_AE_mono) auto

lemma esssup_AE_cong:
"f  borel_measurable M  g  borel_measurable M  AE x in M. f x = g x  esssup M f = esssup M g"
by (auto simp: esssup_def intro!: Limsup_eq)

lemma esssup_const: "emeasure M (space M)  0  esssup M (λx. c) = c"
by (simp add: esssup_def Limsup_const ae_filter_eq_bot_iff)

lemma esssup_cmult: assumes "c > (0::real)" shows "esssup M (λx. c * f x::ereal) = c * esssup M f"
proof -
have "(λx. ereal c * f x)  M M borel  f  M M borel"
proof (subst measurable_cong)
fix ω show "f ω = ereal (1/c) * (ereal c * f ω)"
using 0 < c by (cases "f ω") auto
qed auto
then have "(λx. ereal c * f x)  M M borel  f  M M borel"
by(safe intro!: borel_measurable_ereal_times borel_measurable_const)
with 0<c show ?thesis
by (cases )
(auto simp: esssup_def bot_ereal_def top_ereal_def Limsup_ereal_mult_left)
qed

"esssup M (λx. f x + g x::ereal)  esssup M f + esssup M g"
proof (cases )
case True
then have [measurable]: "(λx. f x + g x)  borel_measurable M" by auto
have "f x + g x  esssup M f + esssup M g" if "f x  esssup M f" "g x  esssup M g" for x
then have "AE x in M. f x + g x  esssup M f + esssup M g"
using esssup_AE[of f M] esssup_AE[of g M] by auto
then show ?thesis using esssup_I by auto
next
case False
then have "esssup M f + esssup M g = " unfolding esssup_def top_ereal_def by auto
then show ?thesis by auto
qed

lemma esssup_zero_space:
"emeasure M (space M) = 0  f  borel_measurable M  esssup M f = (- ::ereal)"
by (simp add: esssup_def ae_filter_eq_bot_iff[symmetric] bot_ereal_def)

end