Limsup of continuous functions pdf

How can we explain limit supremum and limit infimum to. Contents 103 107 109 114 117 127 3 9 145 145 153 157 160 170 17l 175 179 179 189 195 200 204 208 208 216 224 227 232 242 243 250 253. We say that eis bounded above if there exists a real number u. R are borel measurable and functions that are equal a. Let x be a complete separable metic space and b its borel eld. We next show that for discontinuous games, under some mild semicontinuity conditions on the utility functions, it is possible to. Hunter 1 department of mathematics, university of california at davis 1the author was supported in part by the nsf. One can ask, whether there exists a continuous function with such a property. We say that fn converges pointwise to a function f on e for each x.

However, just as there are sets that are not measurable, there are functions that are not measurable. Introduction in order to make us understand the information more on approaches of a given real sequence an n 1, we give two definitions, thier names are upper limit and lower limit. We need only to show that g is measurable since the measurability of h comes. Measurable functions in measure theory are analogous to continuous functions in topology. The superior limit is the larger of the two, and the inferior limit is the smaller of the two. We know simple functions are measurable, and our choice of simple functions approximates fx, therefore fis measurable. It introduces some important tools from analysis which there wasnt time to cover in mas221. Lecture 6 continuous games discontinuous games there are many games in which the utility functions are not continuous e. Limit superior and limit inferior given a sequence a,a,a.

Singular continuous measures by michael pejic 51410. We shall be particularly interested in how general functions can be written as sums of series of simple functions such as power functions and trigonometric functions. If f is increasing and for every n, limn infan, limn supan a,d, then. A continuous function pulls back open sets to open sets, while a measurable function pulls back measurable sets to measurable sets. This will take us to the theories of power series and fourier series. There are various equivalent ways to define lim sup,lim inf. Mx of probability measures converges weakly to a probability measure. A completely identical argument just replace the infs with sups above shows that the limit inferior exists as well. We denote by mx the space of probability measures on x,b. R r is continuous in x, f is sequentially continuous in x. Note, limc n 1 i c n is not bounded below, and limc n 1 i c n is not bounded above. It is a simple exercise to show that e liminf nen limsup en e.

In math202, we study the limit of some sequences, we also see some theorems related to limit. Limsup of continuous functions between metric spaces. In general, we may meet some sequences which does not. In this chapter, we will consider functions from x to ir, where ir. Nevertheless, as the following examples illustrate, it is not as wellbehaved as one might initially expect. Since f is the uniform limit of continuous functions. Chapter 2 measurable functions university of sheffield. Math301 real analysis 2008 fall limit superior and limit. In my view lim sup is 1 and lim inf is 1 for the given. Math 1d, week 2 cauchy sequences, limits superior and inferior, and series3 so the limit superior exists. In this sections, all the subsets of r are assumed to be nonempty. The space d0,1 variable x taking values in s if efxn.

Find two realvalued functions f and g that are uniformly continuous on a set d, but such that their product f g is not uniformly continuous on d. E, the sequence of real numbers fnx converges to the number fx. An extended realvalued function f is upper respectively, lower semicontinuous at a point x 0 if, roughly speaking, the function values for arguments near x 0 are not much higher respectively, lower than fx 0. An illustration of limit superior and limit inferior. R in l1irn by a continuous function of compact support when r is a rectangle. If ff ngis a sequence of measurable functions on x, then. Conversely suppose that we dont know that a n converges but we do know that liminf n. Basic properties of limsup and liminf 1 equivalent definitions. Saying xn converges weakly to x is the same as saying that the laws of. Functionals, duals of normed spaces in this section we consider an important special case of continuous linear maps between normed spaces, namely continuous linear maps from banach spaces to the scalars. The two red curves approach the limit superior and limit inferior of x n, shown as dashed black lines. If a n converges to a limit, then all of its subsequences also converge to the same limit and it follows that lim n. Measurable functions let x be a nonempty set, and let s be a. Mtht 430 chap 8h limsup and liminf for functions see also spivak chapter 8 problem 18 let x k be a bounded sequence.

Let e be a measurable set with positive measure and let a. This idea can be extended to metric spaces such as rd or the space of continuous functions. If f is increasing and for every n, limn infan, limn supan a,d, then lim n supf an f lim n supan and lim n inff an f lim n infan. Theorem lusins let f be a measurable realvalued function on an interval a. Limits and continuous functions mit opencourseware. How can we explain limit supremum and limit infimum to students. Continuous and discontinuous games mit opencourseware. The nal method, of decomposing a function into simple continuous functions, is the simplest, but requires that you have a set of basic continuous functions to start with somewhat akin to using limit rules to nd limits. A function taking values in a countable product of metric spaces is uniformly continuous iff its coordinate functions are. As well, because fa n gis a bounded sequence, the x0sare also bounded. In this case, the sequence accumulates around the two limits. Let fn be a sequence of continuous functions which converge uniformly to a function f on a set e. Efx whenever f is a bounded and continuous function.