Let me now define the integral f of t dWt for any function f from the space L2. This is exactly the second stage in the definition of this kind of integrals. Once more, this integral is referred to as a linear integral. Stage two, f is any function from the space L2, and for the function for the space L2, we can proceed as follows. We can take a sequence fn. This is a sequence of step functions which converges to f in L2 norm. That is, integral from a to b, fn of t minus f of t squared, dt extending to zero as n goes to infinity. And if we found a sequence of such functions, we can define the integral of ft dWt. This I of f as a limit of the corresponding integrals of the functions fn. As it is this is a limit as n goes to infinity of the integral of fn of t dWt from a to b. And this limit as n goes to infinity, should be understood as a mean square sense. That is with medical expectation of I of fn minus I of f squared should go to zero as n goes to infinity. Well, look at this definition. One could ask the following three questions which I would like to address in this lecture. The first question. Why is this definition doesn't depend on the choice of fn? In other words, why if I choose different sequences of step functions which converges to f and then to norm, why I get the same value of integral? Or more simply, one can ask, why the definition is qualitative tool? So, the question is, why I of f does not depend on fn? Second question. Well, when we considered these integrals for step functions we have proven that the integral I of f has a normal distribution with given, with medical expectation equal to zero by the way, and a variance equal to L2 norm as a function f. The question is, is it possible to prove something similar but for f from the space L2 for any function f? So, what is the properties of I of f? And the third question, which is also rather important. Okay. This definition tells us that if fn is a sequences of functions then, the integral is the limit. But why is this sequence exist? This question is not very trivial. So, how to construct fn? So, if there is direct construction scheme of fn. These are three questions and I would like to answer all of them. Let me start with the first question. So, we have the definition. Basically, I of f from this definition depends on the shares of fn, and the question is, whether this definition is coefficient or not? The correctness directly follows from the following theorem. So let fn and fn tilda are two sequences of step functions. We know that both of the sequences converges to f in L2 norm, fn converges to f, and also fn tilda converges to f. The question is that, to show that the limit of I of fn, and the limit of I of fn tilda coincide. Both the limits I understood to the sense of mean squared convergence. Let me prove that this theorem is true. First of all, let me note that I of fn minus I of fn tilda is a difference between two integrals with respect to step functions. And as you can see from the first stage in our construction, this object's integral is a linear function. And therefore, the difference is equal to the integral for the difference fn and fn tilda. Secondly, I'd like to note that the difference between two step functions is also a step function. This also follows directly from the definition of a step function. And therefore, we can use the result, which was shown previously, that this object has a normal distribution with zero mean and variance equal to the L2 norm as a function. Okay, and since we have the difference between these two integrals, is also an integral with respect to step functio., and we know that main value of this integral is equal to zero and the variance is equal to the L2 norm between fn and minus fn tilda. So I'll immediately conclude that this object is equal to the integral from a to b fn of x minus fn tilda of x squared dx. And here, we showed implies a fact is that fn converges to f and L2 norm and fn tilda converges to fn L2 norm. I would like to emphasize simple statement as an exercise, but if we have that these two sequences converges to the same object, then this integral should converge to zero as n goes to infinity. And finally, we'll get that the limits of I of fn and I of fn tilda considered to the mean squared sense as a [inaudible]. So the first theorem, this statement is completely proven, and we have answered the first question. So, we know that this definition doesn't depend on the choice of a step function. Now, let me proceed with the second question. What are the properties of this integral for any f of the space L2? So, for any function f from the space L2, the integral of f has a normal distribution with zero mean and covariance equal to the L2 norm or the function f in the power to- so, integral f squared of x dx integral from a to b. This result can be easily proven. We know is that I of f is defined as a limit and as n goes to infinity of I of fn. And all of these integrals I of fn are integrals of the step functions, and we have shown already as integrals have normal distribution with zero mean and variance equal to the integral fn squared of x dx. Well, there is a result in the probability theory which were used already in when we started the notion of ugadicity, that the limit of normal distribute dependent variables can be also only a normal distribute dependent variable. And also the mathematical expectation of this limit is equal to the limit of- mathematical expectations is equal to the limit of zeroes, and the limit of variance. This is equal to the variance of I of f. So, we conclude finally that I of f have as a normal distribution with zero mean and variance equal to the limit as n goes to infinity integrals a b fn squared of x dx. And this limit is exactly equal to the integral from a to b f squared of x dx because f of n converges to f and L2 norm. And this observation basically concludes the proof. So, we have also asked for a second question is that integral I of f has a normal distribution. And now, we can say that we know integral is normal distributed for any function f from L2. As for the third question, I don't want to make my lectures boring, and I will show you when to construction for even a more general type of integrals, and from that construction, it will directly follow how to proceed for the linear integrals. And the next subsection we will study this more general type of integrals, where we have here not the deterministic function f of t but a stochastic process x of t.