By Marek Kuczma
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 collage in Kraków. He defended his doctoral dissertation below the supervision of Stanislaw Golab. within the 12 months of his habilitation, in 1963, he acquired a place on the Katowice department of the Jagiellonian college (now college of Silesia, Katowice), and labored there until eventually his death.
Besides his numerous administrative positions and his amazing instructing job, he complete 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 e-book describes its contents correctly. most likely even the main dedicated expert do not have concept that approximately three hundred pages could be written with reference to the Cauchy equation (and on a few heavily similar equations and inequalities). And the e-book is under no circumstances chatty, and doesn't even declare completeness. half I lists the necessary initial wisdom in set and degree thought, topology and algebra. half II provides information on options of the Cauchy equation and of the Jensen inequality [...], particularly on non-stop convex services, Hamel bases, on inequalities following from the Jensen inequality [...]. half III bargains with similar 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 sphere of extensions of homomorphisms in general." (Janos Aczel, Mathematical Reviews)
"This e-book is a true vacation for all of the mathematicians independently in their strict speciality. you can actually think what deliciousness represents this e-book for practical equationists." (B. Crstici, Zentralblatt für Mathematik)
Read Online or Download An Introduction to the Theory of Functional Equations and Inequalities: Cauchy's Equation and Jensen's Inequality PDF
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Additional info for An Introduction to the Theory of Functional Equations and Inequalities: Cauchy's Equation and Jensen's Inequality
Proof. nm ∈ Σ , m, n1 , . . nm } . 38 Chapter 2. Topology For the set A choose a set B ∈ Σ such that A ⊂ B and if A ⊂ Z ∈ Σ, then every Y ⊂ B \ Z belongs to Σ. nm , k1 , . . kj ∈ Σ according to the same pattern. nm , n1 , . . nm . nm = B. kj l ∈ Σ , l=1 since Σ is a σ-algebra. kj l ∈ Σ . nm \ z m=0 Let [. nm l . nm l . Then ∞ [. nm \ ∞ ∞ [. ] = m=0 z [. ] . 2) B \ A ∈ Σ. Since A ⊂ B, we have A = B \ (B \ A) ∈ Σ. 2. Every analytic set has the Baire property. In Part II of the present book we will encounter many sets without the Baire property, and hence non-analytic.
Hence Aξ ∩ Aβ ∪ Mβ ⊂ ξ<α Mξ ⊂ Aξ ξ<α δ ∩ ξ<α Mξ σ = Mα ∩ Aα . ξ<α Thus (i) holds in this case, too. (ii) The proof is again by transﬁnite induction with respect to α. For α = 0 (ii) is true in virtue of the well-known property of the open and closed sets. Suppose that for all β < α < Ω we have (∗∗) if Z ∈ Aβ , then Z ∈ Mβ , and conversely. Take a Z ∈ Aα . , for every n ∈ N there exists a ξn < α such ξ<α that En ∈ Mξn . Hence, by (∗∗), En ∈ Aξn , n ∈ N, and ∞ En ∈ Z = n=1 Aξ δ ξ<α The proof of the converse implication is analogous.
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 ) . 9) n=1 ∞ Proof. Fix an ε > 0. We can ﬁnd an open set G such that A ∩ An ⊂ G and n=1 ∞ m(G) − ε me A ∩ An . 2 i=1 me (A ∩ An ) Since ∞ ∞ (G ∩ An ) = G ∩ n=1 me (G ∩ An ) = m(G ∩ An ) . 11) n=1 ∞ me A ∩ ∞ An m(G) − ε (G ∩ An ) − ε m n=1 n=1 ∞ ∞ m(G ∩ An ) − ε = n=1 me (A ∩ An ) − ε . 4. 5 the me (A) = me (A ∩ A1 ) + me (A ∩ A1 ) for all A ⊂ RN , which is equivalent to the measurability of the set A1 , and in the Carath´eodory theory of the Lebesgue measure is even taken as the deﬁnition of the measurability of A1 .
An Introduction to the Theory of Functional Equations and Inequalities: Cauchy's Equation and Jensen's Inequality by Marek Kuczma