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Let (R; M /) be the following (familiar) measure space: if E is countable M= {EeR: either E or R E is countable} _ M(E) = otherwise Describe all bounded integrable ...

Question

Let (R; M /) be the following (familiar) measure space: if E is countable M= {EeR: either E or R E is countable} _ M(E) = otherwise Describe all bounded integrable functions on this measure space and the values of Justify your claims_ Use without proof the description of measurable their integrals functions 0n this kind of space given in the Bonus below. Let (X X 4) be the following measure space: Xis an uncountable set; Bomus. if E is countable I={E €X: either E or X! E is countable}, M(E) =

Let (R; M /) be the following (familiar) measure space: if E is countable M= {EeR: either E or R E is countable} _ M(E) = otherwise Describe all bounded integrable functions on this measure space and the values of Justify your claims_ Use without proof the description of measurable their integrals functions 0n this kind of space given in the Bonus below. Let (X X 4) be the following measure space: Xis an uncountable set; Bomus. if E is countable I={E €X: either E or X! E is countable}, M(E) = otherwise R is measurable, then f is 'essentially constant' i.e. there exist If fx+ number for which f = €. Prove_



Answers

Let $\left\{b_{n}\right\}$ be a sequence and let $a_{n}=b_{n}-b_{n-1} .$ Show that $\sum_{n=1}^{\infty} a_{n}$ converges if and only if $\lim _{n \rightarrow \infty} b_{n}$ exists.

In this problem, we have given a rod of radius art like this and length of this rod is and so this road has no uniform mass density. This is that various row is equal to or not one plus X. Y. L. So first, in the first part we have to sketch a graph between raw and X. So we know that this is equal to Rohnert, X, Y L plus or not. So we will compare it. Why is equal to M X plus C. So we will draw a graph between density and X. So we can say this graph will be like this, this is a graph of pharaoh is equal, do row not one plus X. Y. L. So this is a density at X is equal to zero. If we put in this equation at X is equal to zero. Yeah, at X is equal to zero, density is equal to or not. So this will be equal to Rohnert and at X is equal to L. Suppose this is acceptable to L. The density will be acquittal. If we put yeah X is equal to L. We get 10 city is equal to Rohnert plus row, not into L. Y. L. So this is equal to two times of Rohnert at accessible to add density values. Tour or not. So this is the answer for part A No for part A. We can right had X is equal to zero density is equal to or not at X is equal to L. Density is equal to two times of Rohnert. No second part we have to find the total mass of this rod. Suppose if we see this rod, mm this is like this, it's an X. Axis and this is why access So we will take element and distance X from the origin and width of this element is Bx. So we can write the volume of this element will be equal to bye. R squared because the radius of these roads are so this is fire esquire in two D X. This is the volume so we can write the mass of this element will be equal to rho into V. Or we will read this is D. V. This is very small volume and we know that X distance this density is equal to rho note one plus X. Y. L. And B B is equal to fire square D X. So this is that master of the element. So dear will be equal to run out by our square one plus X. Y. L. Bx. So from here we will integrate it from zero to L. So we can ride total mass is equal to road into fire square. The integration of one plus X Y L. Bx. Soft row fires where we will integrate it. So we can diet this is 02 L B X plus zero two L one by L. X. Dx right? We can write it like this also. So we will right the integration of one dx X. So we will put limits. So this will be called to L plus. This is one by L. And this integration of access access square by two. So this will be called to L square by two. So this is equal to three L helpless. Elway to that mystery away to row by our square. So this is the total mass of the rod. Or we can write it. This is a call to three by two route. Why are the square into L.

Having done all Unicorn Dance Trip room is cool to room. Not one plus ex upon end. So if we plot the rules grab that access called to deal with having that is to do not. Then it starts increasingly nearly like this in second part. If we consider the smaller limit DX. Mhm. At a distance X. The mask of it will be all right ruined. Two dips well not want less X. Upon end in two D. X. To find its total Mazz. Uh huh. We have to integrate it for the limit 02 L. Mhm. Or not one plus X. Upon end. Mhm. The X. Mm. Mhm. So you will get a cronut X plus X squared by to it. For the limit zero to win. So you will get root note and killers and the square going to work minus zero. So on solving it you will get three will not end fight too. They're so thanks for watching it.

Yeah, there's problem. We've been given capital F of X. Is X squared B to the X. In the lower case F of X is either the X times X squared plus two acts. And what we've been passed with showing is that capital F. Because an anti derivative of lower case F. Another way of asking us all is to say show that the derivative of capital F is a little lower case out. This will suffice is here. So let's find the derivative of capital left using our product. All the derivative of X squared is two X two X time is the second, the derivative of the second. So the derivative of E to the X is either the apps times of first exact square. These terms have any into the X. In them. And so we can pull that out. And that will leave this with two X. Los X squared, which is the same as either the X times X squared plus two X. Which is F of X. And so have private access. That's what we want. Mhm.

In this question here were given. Uh, I am equal to the B N minus. B a minus one. And we're interested in the submission that I am from one option infinity. So notice that they will explain this one when any could you want him to be one minus B zero, then plus B two minus, uh, be one plus the B three months, B two plus the big four minus P five plus up to the PM minus P m minus one. And so, um, and we see them, we can console the B one window. Be one here, be to win the B two B 300 b three. So we're missing the beat three years. So sorry. So let me in. Center doesn't before, uh, be 1234 minus b three here. And then we again this one will be canceled with this one and so on. And here we see that we will cancel this and so on on. We see we have left with only, uh if we consider only up to in the past with some s and for now, then we just stop up to here and then we see that this s and we ego Thio actually end here on me and then we see that this sn it will equal to the We have left with the minus p zero then plus with a p n. And we see that this is here as an we turned the limit here Angus t infinity And then he coaches limit on the industry Infinite. They on the minus B plus B n And we say this one we get echoed you This one will be p zero. So it will be the minus p zero and this leap plus the limit on the PM and goes to infinity. I doesn't implies that the I the submission on the I m converge even only if this limit here also. Uh, I already mitt because you some constant else model that infinity


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