Length of a set

[jman...@gmail.com]

Hi,

I have a question about what is happening in this video at 00:32:06 -

http://www.youtube.com/watch?v=99pX1i4bBQY#t=1926s.

Sir in the lecture, you said that we can't define length for any set and it is outside the scope of the lecture. Can we define it for only those sets which have denseness property and are subset of a complete ordered field? I am not sure whether I am correct or not. Can you give example where we can't define the length and explain a little bit?.

Thanks!

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Reference Key - Key('StudentQuestionEntity', 5564652355321856, namespace='ns_noc21_ma31')

[Vinay M]

Good question, the difficulty here arises from sample space which here is an interval, then how to define an event and distribute probability, because there are uncountable number of points in the sample space. Are each points an event? If so then there are infinite of them, but the total probability of sample space should be 1. So now does that imply probability of a single point as zero? But if we again sum those we'll get total zero, a contradiction. To overcome this obstacle we have to introduce a notion of length being the factor which decides probability and when we say length it means distance metric in R of a continuous interval or unions of piecewise continuous intervals. Only then "length" notion has meaning. Now what happens if we take a set with no intervals but some isolated points or even let's say a subset of irrational numbers? Then there is no interval here and therefore cannot use the length notion to get probabilities. This is my understanding, I could be wrong. Thanks for the question. 

[Himanshu]

@Vinay thaks for the reply.   

I also think so. Because irrational number does not obey completeness property. Because if we define an event to be (1/2,1) having all irrational numbers, then no supremum or infimum exists in set of irrational number though it is bounded.    

But in week-7 lecture-2, at 6:18, sir said that we can define length of any set A by considering it to be in Borel sigma field. 

So, in the above example if we consider Borel sigma field to be {(0,1/2),(1/2,1),(0,1)} (with all intervals having irrational numbers), with the above result we can define length to (1/2,1), though it has no rational numbers.    

I don't know how to define length in that case. That is my question.   

Because inttuitively we think that length is defined for intervals which are subset of a complete ordered field.   

But in the above case of irrational numbers I don't understand that.   

Thanks!

[Vinay M]

I think at 6:18 of Lec 2 of week 7, he said that if we consider the special type of Sigma field called Borel Sigma field as our event space, then we can define length for any element of this event space. Since "Borel Sigma field" is a type of Sigma field that is the smallest sigma field which has all the intervals of the sample space. This implies that the Borel Sigma field must only contain intervals, don't you think? But I too have doubts here, it is, since a single point is also considered as an interval of length zero, then the Borel Sigma field should also contain, single points as events, now if we take a bunch of single point events and want the probability of their unions, then it is zero and we go back to same obstacle. 

What you were thinking, that we can define event space for any sigma field was mentioned by professor at 17:20. I need to look up some of these. Maybe if TA replies, it helps. Thank you for the question and also the replies. 

[Himanshu]

@Vinay, I think we have to consider just intervals in the borel sigma field as also mentioned in the lectures. But I think that we can define event space for any sigma field. I don't know how. 

Yes, TA's reply will be very helpful.