What is the smallest sigma-algebra that contains the set of singleton sets?

[ss1112]

1- Give an example for sequence En measurable such that
m* (∩En)<lim m*(En).
2- find smallset sigma-algabra contains the set {{x}:x in R }
3- prove that if fn convergence almost everywhere to f then f is measurable.
4- prove that decreasing function F is measurable or given example if F is
not measurable .

 

[Bacle]

Where are you stuck?

 

[ss1112]

thanks ,
In all of these questions

 

[Bacle]

I don't mean to drag you along, but, do you know the definitions?
i.e., how do we calculate the outer measure m*, and, what is a sigma algebra, and
some properties of decreasing functions . Do you know the meaning of almosy everywhere?
Do you know when we define a function to be measurable?

You need to know , or at least have a good idea of these definitions, to be
able to answer these questions. I mean,e.g., if I told you that the lim sup of a
sequence of measurable functions is measurable and that when a sequence
converges, the limit equals the lim sup, or that a decreasing (monotone) function
is a.e. differentiable (and what can we conclude from differentiability of f?)
I imagine would not help much.

Go over the definitions and tell us if/where you're stuck,
and we'll help you through. Good luck.

 

[ss1112]

Know the possible definitions of reading any book in the real analysis

 

[gb7nash]

Know the possible definitions of reading any book in the real analysis

Is that a question or a statement?

If you want to get help with this problem, you need to put forth more effort than this. What definitions do you know? How far did you get? Where are you stuck? etc. If you don't know the definitions at all, this is a problem you need to ask your instructor.

 

[ss1112]

thanks