 There is another way of extending the integral. So, let us look at that. What is that way? So, a Riemann integral on AB to R bounded. So, we defined with respect to a partition P, the lower sum and the upper sums. They were all less than or equal to this. And as the, you took the supremum. So, this was, you look at the supremum of L P f is less than or equal to infimum of U P f. And whenever these two were equal, that was called the integral of A to B f x B x. But the important thing in this was the, in the upper sum or the lower sum, say for example, in the lower sum, if the partition P was x 0 less than x 1 less than x n equal to V, then you looked at the lowest value. You are approximating by looking at the lowest height of the function into the length of the interval. So, the idea was that you look at the length of that sub interval. At some time point, somebody thought, so what we are doing is, you are given an interval A to B on the line and you are trying to measure the length of that interval. You measure the length by looking at, saying that this is an interval B minus A. So, length is B minus A. So, that is the length of this interval. This is I. But I think probably that may be a nice idea to introduce. So, length on the class of all intervals taking value 0 to infinity. So, length of an interval I is defined as B minus A, if n points A and B. Otherwise, you write as plus infinity. If it is an embodied interval, the length is defined as plus infinity. But this is something I can change this notion of length. So, what I am here, when the n point is A and B, I am saying it is B minus A. Let us take alpha function on real line to real line, which is monotonically increasing. So, let us take a function. Then what is wrong in saying, I define the length of interval I to be equal to alpha B minus alpha A, if I has n points A and B. What is wrong in saying that, if I have the length to be equal to alpha B minus alpha A. That means, at every point, I am giving the different weightage to every point, where the earlier the length was B minus A. Now, the length is alpha B minus. If alpha is equal to identity function, then it is the original length. So, the idea came that, when defining that upper sum and the lower sums, UPF, let us define the upper sum with respect to a new length function alpha. So, what is going to be the definition? It is the maximum value of the function as it is. The height remains the same, but the base length changes to alpha x i minus alpha of x i minus 1 and summation i equal to 1 to n. Change the length of intervals and similarly, the lower sum with respect to this alpha, I can define it as sigma m i alpha x i minus alpha of x i minus 1. Still, the same property remains true. LPF alpha is still less than UPF alpha. Is that okay? Because these two quantities are the same, the weightage of the length is same, small m i and capital M i. So, that still remains the same. So, still I can ask, what will happen if I refine a partition? Because if you keep in mind, in the lower sums and the upper sums, when you refine a partition, length does not play any part. It is a supremum or the infremum that changes. So, same properties of lower sums and upper sums will remain true whenever you measure the length in terms of a monotonically increasing right continuous function, monotonically increasing function. So, this, so you can look at what is the supremum of lower sums, what is the infremum of upper sums and whenever they are equal, you will get a new integral which is defined with respect to a weighted measurement of length on the line. So, this is possible and this is what is called Riemann still j's integral. So, let me, because not many changes come except for a few places, otherwise the whole theory goes as smoothly as the earlier one. So, that is what we are, I think. So, let us start with a function A B to R and alpha is a monotonically increasing function on A B. Given a partition, look at the infremum, look at the supremum like we do it for upper sum and lower sums. Now, how is the new upper sum defined? How is the new lower sum? Minimum value into alpha xi minus alpha xi minus 1. So, lower sum, similarly the upper sum with respect to the, is a weight you are attaching to each. See, why this is important in probability and statistics? You assign different weights to different points. So, you will have probability distributions which come via such kind of functions, mononon functions and the expected value of the functions will be with respect to the weighted measurement of length of that interval. So, that is why this is important from statistics point of view and mathematics point of view, it is a generalization of the Riemann integral. So, upper sum and lower sum, so you prove those property that the upper lower sum is always less than or equal to upper sum. That is not a difficult job. There is a same thing, same proof continues. You can define upper Riemann still j integral, lower Riemann as a supremum and the infremum of the upper and you say the function is Riemann still j integrable. The supremum of the lower sums with respect to alpha is equal to the infremum of the upper sums with respect to alpha. All everything goes on smoothly. So, you say that is the integral and common value is called the integral. Probably all the theorems go as in Riemann integral except probably some points. I will point out where the things go a bit. So, for example, upper is always lower than the bigger than lower and the function is integrable whenever the difference can be made small because that is the only thing that makes a difference. Alpha length does not contribute anything actually. So, all those theorems, same proofs go over. Only difference in the proofs will be that instead of writing x i minus x i minus 1, we will be writing alpha x i minus alpha x i minus 1. That is all, nothing more. So, all those proofs go. So, here is something. If every continuous function is Riemann still j integrable like in Riemann integral, we proved that if f is continuous on a, b then it is integrable. So, what was the proof basically? The proof basically was that because f is defined on a close bounded interval, it is continuous. So, it is uniformly continuous. So, given epsilon, there is a delta, so that whenever two points are close by distance delta, their distance f of x i, the values are also close. Now, what are the upper sums and lower sums? For a partition, you look at the points upper sum minus the lower sum, capital M i minus small m i. What is that? If the function is continuous, capital M i is attained at some point, small m i is attained at some point. That means, in the interval x i minus 1 to x i, capital M i is f of something in that interval, small m i is f of something. So, if your length of the partition is less than delta, then the values will be less than epsilon because of uniform continuity. So, given f is uniform, f is continuous by uniform continuity, given epsilon, choose a delta, choose a partition whose norm is less than delta. So, you will get capital M i minus small m i will be less than epsilon into the length, so that will be small. So, that was the proof. So, in the same proof, if f is continuous, uniform continuity, everything makes sense. So, what we will be doing? Instead of multiplying by x i minus 1 to x i, we will be multiplying by alpha times x i minus alpha times x i minus 1. Same proof, no change will come. So, every continuous function will also be Riemann Stelge's integrable with respect to alpha for the same proof, if f is continuous. No change comes. The change comes supposing f is not continuous, it is only a bounded function and alpha is monotonically increasing. We know that for a bounded function Riemann integral lean not exists. So, the modified theorem for Riemann Stelge's integral is, if that function alpha, which is monotonically increasing, is continuous, every discontinuity point of f, then it becomes Riemann Stelge integrable. So, the discontinuity of f is taken care by the continuity of the point, monotonically increasing function. Alpha is monotonically increasing. So, it is discontinuous at the most, at countably many points. We know that. So, the only change comes with, it is continuous at every point of discontinuity of f. Then, so that is for any bounded for Riemann integrable function, we said that every monoton function is Riemann integrable. For Riemann Stelge, you require f to be, alpha to be a continuous. Not only monotonically increasing, you require also it to be a continuous function. We will not prove all these theorems, but I am just pointing out the differences between the two statements of the theorems. So, that is properties of Riemann Stelge. Other facts, they all continue to hold. If f and g are Riemann Stelge integrable, f plus g is Riemann Stelge integrable, because the sums will split anyway into two parts. So, length does not change those properties. So, Riemann integrable of f plus g is Riemann integrable of f plus Riemann integrable of g, same property holds. Scalar multiple integrable comes out, f less than g, then same property holds. So, similar properties hold, not much change comes. So, that is why not, proofs are not very interesting for all this, but we will not anyway go into the proofs of these things. So, the basic fact is that you can extend Riemann integral in two different ways. One, look at function being unbounded on a bounded domain or function being bounded on an unbounded interval. That gives you one set, which is called improper integration. Or, you can keep the function values as it is, but change the measurement of length by some monotonically increasing function. That gives you Riemann Stelge's integration. The two branches, which are, I think, probably I should just say that this Riemann Stelge integral plays an important part in further topics like measure theory and probability and statistics. It will come again and again there. And, of course, improper integral comes in a disguised form of gamma functions and gamma function and Cauchy distributions and so on or normal distributions and so on. So, you will require those things to exist. So, that is integration of one variable. So, next time we will look at integration of functions of two variables. That is also important from many subjects quite a few. So, we will look at integration of functions of several variables next time.