The rest is from hoveys book model categories hov99. The limits of the infimum and supremum of parts of sequences of real numbers are used in some convergence tests. Show that for every 0 there exists a set eof measure e such that f n converges uniformly to foutside the set e. Relation between lower and upper lim its the lower and upper limits of a set sequence an satisfy liman.
Let s denote the set of all real numbers for which there exists at least one subsequence. Then by what we have already proved, there is a subsequence y n k of y n converging to limsupy n, and then x n k. They are extensively used in real analysis, including the axiomatic construction of the real numbers and the formal definition of the riemann integral. We have the following interpretation for the lim inf and lim sup limits. Informally, for a sequence in r, the limit superior, or limsup, of a sequence is the largest subsequential limit. Set limits, particularly the limit infimum and the limit supremum, are essential for probability and measure theory. In this case, the sequence accumulates around the two limits. Let e be a subset of rn, and inte the set of all interior points of e. Set theory this chapter describes set theory, a mathematical theory that underlies all of modern mathematics. Real analysis spring 2003 solutions to some problems. But even more, set theory is the milieu in which mathematics takes place today. Some sets are bigger than others the cardinality of a set. General measures both of the above examples are accommodated by a generalization.
K represents the characteristic function of the set k. Limit superior and limit inferior given a sequence a,a,a. The two red curves approach the limit superior and limit inferior of x n, shown as dashed black lines. Measure theoretic laws for lim sup sets memoirs of the. In general, nding a limit of a sequence of sets is not easy and we will not do this here. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. This chapter describes set theory, a mathematical theory that underlies all of. Math 280 probability theory lecture notes ucsd mathematics. Sets may be described by listing their elements between curly braces, for example 1,2,3 is the set containing the elements 1, 2, and 3. We establish sufficient conditions which are also necessary under some natural assumptions for the \m\measure of \\lambda\psi\ to be either positive or full in \\omega\ and for the.
Hence ais the union of an open set, inta, and a subset of the null set. Set theory plays a central role in the theory of probability. The classical lim sup set \w\psi\ of \\psi\approximable numbers in the theory of metric diophantine approximation fall within this class. Math301 real analysis 2008 fall limit superior and limit. Basic set theory a set is a many that allows itself to be thought of as a one. For those of you new to abstract mathematics elementary does not mean simple though much of the material. Set theory, and set operations statistical science. Buy measure theoretic laws for lim sup sets memoirs of the american mathematical society on free shipping on qualified orders measure theoretic laws for lim sup sets memoirs of the american mathematical society. The lower limit, liminf xn is the set of points which belong to xn for all but a. A comprehensive collection of rules for set calculus. Show lim sup an lim inf an for increasing for increasing sequence of subsets.
The infimum and supremum are concepts in mathematical analysis that generalize the notions of minimum and maximum of finite sets. B for example, if s is the set of all juices in the supermarket, and t is the set of all. Hitting probabilities of random covering sets in tori and metric spaces. This article is restricted to that situation as it is the only one relevant for measure theory and probability. Observe that for each x, each term in this series is either 1 or 0. The sequence a n is bounded and increasing, so it has a limit.
Set theory if we are interested in elements of a set a that are not contained in a set b, we can write this set as a. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. Then by the axiom schema of comprehension, there is a set bdf x2vw g. To understand why the theorem is likely to be correct not proving it, of course, one can just try constructing a counterexample, a. Lim inf and lim sup 3 we have shown that limsupx n is the largest limit of convergent subsequences of x n. Victor beresnevich, detta dickinson, and sanju velani. The superior limit is the larger of the two, and the inferior limit is the smaller of the two. Since the latter is always measurable, we conclude that ais a measurable set.
An introduction to set theory university of toronto. An illustration of limit superior and limit inferior. Theorem egoro s if f n is a sequence of measurable functions that converge to a realvalued. Basic properties of limsup and liminf 1 equivalent definitions aau.
Measure theory is not needed unless we discuss the probability of two types of events. If you try to guess the answer quickly, you might get confused between an ordinary supremum and the lim sup, or the regular infimum and the lim inf. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. Probabilities are assigned by a pa to ain a subset f of all. Furthermore, we might be interested in more complicated rare events, beyond the interval a. In mathematics, the notion of a set is a primitive notion. But we know that the integral of gis equal to the sum in. The compactness conditions for d 2 are considerably more cumbersome, see 4, 93. Thus f kg k 1 converges towards the in mum of its range. A n is not the set of accumulation points, which would be the entire interval 0, 1 according to the usual euclidean metric. This alone assures the subject of a place prominent in human culture. The upper limit, lim sup xn of a sequence xn of sets is the set of points which belong to xn for in. They are not guaranteed to be comprehensive of the material covered in the course.
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