src/HOL/Analysis/Bochner_Integration.thy

   moreover have "integrable M (\<lambda>x. min (f j x) (Min {f i x |i. i \<in> F}))"
     by (rule integrable_min, auto simp add: H)
   ultimately show "integrable M (\<lambda>x. Min {f i x |i. i = j \<or> i \<in> F})" by simp
 qed
 
+lemma integrable_MIN:
+  fixes f:: "_ \<Rightarrow> _ \<Rightarrow> real"
+  assumes "finite I" "I \<noteq> {}"
+          "\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)"
+  shows "integrable M (\<lambda>x. MIN i\<in>I. f i x)"
+using integrable_Min[of I M f, OF assms]
+by(simp add: Setcompr_eq_image)
+
 lemma integrable_Max:
   fixes f:: "_ \<Rightarrow> _ \<Rightarrow> real"
   assumes "finite I" "I \<noteq> {}"
           "\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)"
   shows "integrable M (\<lambda>x. Max {f i x|i. i \<in> I})"

     using H(1) H(2) Max_insert by simp
   moreover have "integrable M (\<lambda>x. max (f j x) (Max {f i x |i. i \<in> F}))"
     by (rule integrable_max, auto simp add: H)
   ultimately show "integrable M (\<lambda>x. Max {f i x |i. i = j \<or> i \<in> F})" by simp
 qed
+
+lemma integrable_MAX:
+  fixes f:: "_ \<Rightarrow> _ \<Rightarrow> real"
+  assumes "finite I" "I \<noteq> {}"
+          "\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)"
+  shows "integrable M (\<lambda>x. MAX i\<in>I. f i x)"
+using integrable_Max[of I M f, OF assms]
+by(simp add: Setcompr_eq_image)
 
 lemma integral_Markov_inequality:
   assumes [measurable]: "integrable M u" and "AE x in M. 0 \<le> u x" "0 < (c::real)"
   shows "(emeasure M) {x\<in>space M. u x \<ge> c} \<le> (1/c) * (\<integral>x. u x \<partial>M)"
 proof -