By Marek Kuczma

Marek Kuczma was once born in 1935 in Katowice, Poland, and died there in 1991.

After completing highschool in his domestic city, he studied on the Jagiellonian college in Krak?w. He defended his doctoral dissertation less than the supervision of Stanislaw Golab. within the yr of his habilitation, in 1963, he acquired a place on the Katowice department of the Jagiellonian collage (now college of Silesia, Katowice), and labored there until his death.

Besides his a number of administrative positions and his impressive educating job, he entire very good and wealthy medical paintings publishing 3 monographs and one hundred eighty clinical papers.

He is taken into account to be the founding father of the distinguished Polish university of practical equations and inequalities.

"The moment 1/2 the identify of this e-book describes its contents thoroughly. most likely even the main dedicated professional wouldn't have proposal that approximately three hundred pages will be written as regards to the Cauchy equation (and on a few heavily comparable equations and inequalities). And the booklet is in no way chatty, and doesn't even declare completeness. half I lists the mandatory initial wisdom in set and degree idea, topology and algebra. half II supplies information on suggestions of the Cauchy equation and of the Jensen inequality [...], specifically on non-stop convex capabilities, Hamel bases, on inequalities following from the Jensen inequality [...]. half III offers with similar equations and inequalities (in specific, Pexider, Hossz?, and conditional equations, derivations, convex features of upper order, subadditive services and balance theorems). It concludes with an expedition into the sphere of extensions of homomorphisms in general." (Janos Aczel, Mathematical Reviews)

"This publication is a true vacation for all of the mathematicians independently in their strict speciality. you can think what deliciousness represents this ebook for practical equationists." (B. Crstici, Zentralblatt f?r Mathematik)

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4. If An ⊂ RN , n ∈ N, are arbitrary sets, and A ⊂ ∞ An , then n=1 ∞ me (A) me (An ) . 5) n=1 In particular ∞ me ∞ An me (An ) . 6) n=1 Proof. Fix an ε > 0. 1), for every n ∈ N we can ﬁnd an open set Gn such that An ⊂ Gn , m(Gn ) ∞ me (An ) + ε . 7) Gn also is an open set, and hence measurable. 6) are true. 7). 5). 5). 7) are true also for ﬁnite unions. It is enough to put An = ∅ for n > m. 5) the monotonicity of the outer measure: If A ⊂ B, then me (A) me (B) . The Lebesgue measure, as is well known, is σ-additive: if An ∈ L , n ∈ N, are pairwise disjoint: Ai ∩ Aj = ∅ for i = j , i, j ∈ N , then ∞ m ∞ An = n=1 m(An ) .

The σ-algebra of all Lebesgue measurable subsets of RN is denoted by L. ). We do not presuppose any of those approaches. At any case the reader should be familiar with the following theorem, which we quote here without proof. 1. For any set A ⊂ RN the following conditions are equivalent: (i) A ∈ L. (ii) For every ε > 0 there exist open sets G and U such that A ⊂ G , G \ A ⊂ U , m(U ) < ε . (iii) There exist sets H ∈ Gδ and B ∈ L such that A ⊂ H , A = H \ B , m(B) = 0 . (iv) For every ε > 0 there exist a closed set F and an open set V such that F ⊂ A , A \ F ⊂ V , m(V ) < ε .

Let F be a set that has exactly one element in common with every B ∈ B. If B ∩ A × {A} = x, {A} , put w(A) = x. This formula deﬁnes a choice function w : A → A. P. J. Cohen [48] proved that the Axiom of Choice is independent of the remaining Axioms 1–8. A detailed discussion of The Axiom of Choice is found in Jech [154]. Exercises 1. Prove that for any two sets A, B there exists the set A ∩ B containing those and only those elements which are common to A and B. 2. The ordered pair a, b is deﬁned as {a, {a, b}}, where {a, b} is the unordered pair consisting of a and b.