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.