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 collage in Kraków. He defended his doctoral dissertation less than the supervision of Stanislaw Golab. within the yr of his habilitation, in 1963, he bought a place on the Katowice department of the Jagiellonian college (now college of Silesia, Katowice), and labored there until his death.

Besides his numerous administrative positions and his amazing educating task, he finished very good and wealthy medical paintings publishing 3 monographs and a hundred and eighty clinical papers.

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

"The moment half the identify of this booklet describes its contents competently. most likely even the main dedicated expert wouldn't have notion 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 booklet is on no account chatty, and doesn't even declare completeness. half I lists the mandatory initial wisdom in set and degree thought, topology and algebra. half II provides information on strategies of the Cauchy equation and of the Jensen inequality [...], particularly on non-stop convex capabilities, 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 services of upper order, subadditive services and balance theorems). It concludes with an day trip into the sector of extensions of homomorphisms in general." (Janos Aczel, Mathematical Reviews)

"This booklet is a true vacation for all of the mathematicians independently in their strict speciality. one could think what deliciousness represents this ebook for useful equationists." (B. Crstici, Zentralblatt für Mathematik)

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**Example text**

2. 8. Let Y be a separable topological space, and let A ⊂ X, B ⊂ Y . Then the set A × B ⊂ X × Y is of the ﬁrst category if and only if at least one of the sets A, B is of the ﬁrst category. Proof. Suppose that A × B is of the ﬁrst category, and B is of the second category. 7 there exists a set P ⊂ X, of the ﬁrst category, such that for every x ∈ P the set (A × B)[x] is of the ﬁrst category. But for x ∈ A the set (A × B)[x] = B is of the second category. Hence A ⊂ P , and thus A is of the ﬁrst category.

Hence A ⊂ P , and thus A is of the ﬁrst category. To prove the converse implication, we ﬁrst prove that if A is nowhere dense in X, then A× Y is nowhere dense in X × Y . We have cl(A× Y ) = cl A× cl Y = cl A× Y , whence int cl(A × Y ) = int(cl A × Y ) = int cl A × int Y = ∅ × Y = ∅. Thus A × Y is nowhere dense. ∞ Now suppose that the set A is of the ﬁrst category. Then A = ∞ are nowhere dense, whence A × B ⊂ A × Y = An × Y = n=1 ∞ set An × Y, n ∈ N, is nowhere dense, whence ∞ An , where An n=1 (An × Y ) .

V) There exist sets E ∈ Fσ and C ∈ L such that E ⊂ A , A = E ∪ C , m(C) = 0 . 1 will be our main tool in the sequel of this chapter. 48 Chapter 3. Measure Theory Let A ⊂ RN be an arbitrary set. We deﬁne the outer measure1 and the inner measure of A (denoted in the sequel by me (A) and mi (A), respectively) as me (A) = inf m(G), mi (A) = G⊃A G open sup m(F ) . 1) The outer and inner measure are deﬁned for all sets A ⊂ RN . 1) that me (∅) = mi (∅) = 0 . 2. If A ∈ L, then me (A) = mi (A) = m(A) .