# An Introduction to Number Theory by Harold M. Stark

By Harold M. Stark

Nearly all of scholars who take classes in quantity concept are arithmetic majors who won't turn into quantity theorists. a lot of them will, even though, train arithmetic on the highschool or junior university point, and this publication is meant for these scholars studying to coach, as well as a cautious presentation of the traditional fabric often taught in a primary path in user-friendly quantity thought, this booklet features a bankruptcy on quadratic fields which the writer has designed to make scholars take into consideration many of the "obvious" ideas they've got taken with no consideration prior. The ebook additionally incorporates a huge variety of workouts, lots of that are nonstandard.

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N. 2 (Riesz). If a sequence f,, (f,,E V ) converges in norm to f, then f E U and there is a subsequence which converges to f almost everywhere. Proof. We have Jl f m - f, l S Jlfm - fl +Jlf, - fl. By hypothesis, both the integrals on the right side tend to 0, as m + m and n+m. Hence lim m-m Ifm -f, , l = 0, which means that f,, is a Cauchy sequence. Since the space 0 is complete, by Theorem 4,Chapter 11. n. with gE U. , we have f g, which implies that f E U. n. , If,, I + If&- f P , I + If, - fhI + * *

N h is integrable, by the Lebesgue theorem. This means that the sum f + g is measurable, and the proof is complete. 5. The modulus of a measurable function is measurable. Proof. n gl. 6. The intersection and the union of two measurable functions are measurable. Proof. 4. 7. The retract of a measurable function by another measurable function is measurable. n g is measurable. Proof. nh ) so is. n h, which proves our assertion. VI. 8. I f f and g are measurable functions and one of them is real f g is measurable.

Chapter V Applications to Set Theory 1. Characteristic functions of sets Given any subset 2 of K, one often denotes by xz or by l Z the function whose value is 1 at every point of 2, and 0 at the remaining points of K. This function is called the characteristic function of the set Z. A function f (defined on K ) which admits values 0 and 1 only will be called a zero-one function. There is a one-to-one correspondence between sets and zero-one functions. It is easy to list relations for sets and the corresponding relations for zero-one functions.