Limsup of continuous functions pdf

Let x be a complete separable metic space and b its borel eld. As a consequence of the above theorem, we see that a continuous function from ir to ir is lebesgue measurable. Limsup of continuous functions between metric spaces. In math202, we study the limit of some sequences, we also see some theorems related to limit. If a n converges to a limit, then all of its subsequences also converge to the same limit and it follows that lim n. In my view lim sup is 1 and lim inf is 1 for the given. Limits and continuous functions limits of y x are not the only limits in mathematics. However, just as there are sets that are not measurable, there are functions that are not measurable. We denote by mx the space of probability measures on x,b.

We next show that for discontinuous games, under some mild semicontinuity conditions on the utility functions, it is possible to. Efx whenever f is a bounded and continuous function. R r is continuous in x, f is sequentially continuous in x. How can we explain limit supremum and limit infimum to. Let fn be a sequence of continuous functions which converge uniformly to a function f on a set e. Limit superior and limit inferior given a sequence a,a,a. Mtht 430 chap 8h limsup and liminf for functions see also spivak chapter 8 problem 18 let x k be a bounded sequence. In particular, the supremum of a family of continuous functions is l. 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. Basic properties of limsup and liminf 1 equivalent definitions. Math 1d, week 2 cauchy sequences, limits superior and inferior, and series3 so the limit superior exists. 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 define the limit superior and limit inferior of the sequence to be.

There are various equivalent ways to define lim sup,lim inf. Theorem lusins let f be a measurable realvalued function on an interval a. It introduces some important tools from analysis which there wasnt time to cover in mas221. 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. It is a simple exercise to show that e liminf nen limsup en e. A function taking values in a countable product of metric spaces is uniformly continuous iff its coordinate functions are. 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. In general, we may meet some sequences which does not. Hunter 1 department of mathematics, university of california at davis 1the author was supported in part by the nsf. For example the sequence whose nth term is 1n fails to. Saying xn converges weakly to x is the same as saying that the laws of. We say that fn converges pointwise to a function f on e for each x. Let e be a measurable set with positive measure and let a.

Since f is the uniform limit of continuous functions. Measurable functions let x be a nonempty set, and let s be a. Mx of probability measures converges weakly to a probability measure. We say that eis bounded above if there exists a real number u. In mathematical analysis, semicontinuity or semicontinuity is a property of extended realvalued functions that is weaker than continuity. We know simple functions are measurable, and our choice of simple functions approximates fx, therefore fis measurable. Singular continuous measures by michael pejic 51410. If f is increasing and for every n, limn infan, limn supan a,d, then. 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. One can ask, whether there exists a continuous function with such a property. Chapter 2 measurable functions university of sheffield. How can we explain limit supremum and limit infimum to students. 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.

Nevertheless, as the following examples illustrate, it is not as wellbehaved as one might initially expect. An illustration of limit superior and limit inferior. Note, limc n 1 i c n is not bounded below, and limc n 1 i c n is not bounded above. 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. The space d0,1 variable x taking values in s if efxn.

This will take us to the theories of power series and fourier series. Limits and continuous functions mit opencourseware. In this sections, all the subsets of r are assumed to be nonempty. In the following, we will consider extended real number system. Note that from the above definition both lim inf and lim sup must exist. E, the sequence of real numbers fnx converges to the number fx. We need only to show that g is measurable since the measurability of h comes. Math301 real analysis 2008 fall limit superior and limit. 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. In this case, the sequence accumulates around the two limits.

Measurable functions in measure theory are analogous to continuous functions in topology. If ff ngis a sequence of measurable functions on x, then. This idea can be extended to metric spaces such as rd or the space of continuous functions. Singular continuous measures by michael pejic 51410 preliminaries given a set x, a. In this chapter, we will consider functions from x to ir, where ir.

The superior limit is the larger of the two, and the inferior limit is the smaller of the two. A continuous function pulls back open sets to open sets, while a measurable function pulls back measurable sets to measurable sets. The two red curves approach the limit superior and limit inferior of x n, shown as dashed black lines. A completely identical argument just replace the infs with sups above shows that the limit inferior exists as well. R in l1irn by a continuous function of compact support when r is a rectangle. As well, because fa n gis a bounded sequence, the x0sare also bounded.