By Marek Kuczma (auth.), Attila Gilányi (eds.)

Marek Kuczma used to be 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 Kraków. He defended his doctoral dissertation less than the supervision of Stanislaw Golab. within the 12 months of his habilitation, in 1963, he received a place on the Katowice department of the Jagiellonian collage (now collage of Silesia, Katowice), and labored there until eventually his death.

Besides his a number of administrative positions and his impressive instructing task, he entire first-class and wealthy clinical paintings publishing 3 monographs and a hundred and eighty medical papers.

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

"The moment 1/2 the name of this booklet describes its contents accurately. most likely even the main dedicated professional should not have inspiration that approximately three hundred pages will be written with regards to the Cauchy equation (and on a few heavily similar equations and inequalities). And the booklet is under no circumstances chatty, and doesn't even declare completeness. half I lists the necessary initial wisdom in set and degree conception, topology and algebra. half II supplies information on options of the Cauchy equation and of the Jensen inequality [...], specifically on non-stop convex services, Hamel bases, on inequalities following from the Jensen inequality [...]. half III offers with comparable equations and inequalities (in specific, Pexider, Hosszú, and conditional equations, derivations, convex features of upper order, subadditive features and balance theorems). It concludes with an day trip into the sector of extensions of homomorphisms in general." (Janos Aczel, Mathematical Reviews)

"This e-book is a true vacation for the entire mathematicians independently in their strict speciality. you possibly can think what deliciousness represents this ebook for practical equationists." (B. Crstici, Zentralblatt für Mathematik)

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**Additional info for An Introduction to the Theory of Functional Equations and Inequalities: Cauchy’s Equation and Jensen’s Inequality**

**Example text**

2. 1) coincides with the previous notion. 1. Outer and inner measure 49 The next four theorems give the relation between the outer [inner] measure of the union of a countable collection of sets and those of particular sets. 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.

Since F ⊂ F ⊂F∩ ∞ An = n=1 ∞ ∞ An , also n=1 (F ∩ An ). Moreover, we have F ∩ An ⊂ A ∩ An , n ∈ N. Hence n=1 by the monotonicity of the inner measure ∞ m(F ) m ∞ (F ∩ An ) = n=1 ∞ m(F ∩ An ) = n=1 ∞ mi (F ∩ An ) n=1 mi (A ∩ An ) . 19) imply ∞ mi (A ∩ An ) . a< n=1 52 Chapter 3. 17). Before proceeding further let us note that for an arbitrary set A ⊂ RN we have mi (A) me (A) . 1), but one can also argue as follows: For an arbitrary ε > 0 there exists a closed set F ⊂ A such that mi (A) m(F ) + ε .

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 ) . n=1 For the outer measure we have the following weaker version of this fact. 50 Chapter 3. 5. If An ∈ L , n ∈ N, are pairwise disjoint measurable sets, and A ⊂ RN is arbitrary, then ∞ ∞ me A ∩ me (A ∩ An ) .