Why Isn't A^c the Empty Set in This Measure Theory Example?

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Discussion Overview

The discussion revolves around a measure theory example from "Probability and Measure" by Patrick Billingsley, specifically addressing the complement of a set A within the interval Ω = (0,1]. Participants are trying to understand why the complement A^c is not the empty set and how it is expressed in the text.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions how A^c can equal a union of intervals when A is defined as a collection of intervals, suggesting that A^c should be empty.
  • Another participant clarifies that if A is expressed as a union of intervals, then A^c should indeed be empty if A covers the entire interval (0,1].
  • Some participants express confusion over the notation and meaning of A, with one suggesting that A should be interpreted as a union of intervals rather than a set of intervals.
  • There are differing interpretations of the notation (0,0] and whether it is empty or contains the point 0, leading to further confusion about the complement.
  • Several participants attempt to reconcile the definitions and expressions given in the text with their own interpretations of set A and its complement.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the interpretation of A and its complement. There are multiple competing views regarding the definitions and implications of the sets involved, particularly concerning the nature of (0,0] and whether it is empty.

Contextual Notes

There are unresolved assumptions regarding the definitions of the sets and intervals involved, particularly the interpretation of the notation used in the example. The discussion reflects varying levels of understanding of measure theory concepts.

woundedtiger4
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Hi all,
I am reading Probability and Measure by Patrick Billingsley, and I am stuck at one example, please help me understanding it.
http://desmond.imageshack.us/Himg201/scaled.php?server=201&filename=30935274.jpg&res=landing
Ω=(0,1]
My question is that how come the A^c = (0,a_1]U(a'_1, a_2]U...U(a'_m-1, a_m]U(a'_m, 1] ? because let's say that A= {(0,0.1], (0.2, 0.3], (0.4, 0.5], (0.6, 0.7], (0.8, 1]} then
A^c = Ω - A
A^c = (0, 1] - {(0,0.1], (0.2, 0.3], (0.4, 0.5], (0.6, 0.7], (0.8, 1]}
A^c = ∅ ...an empty set?

You can see this example at http://books.google.co.uk/books?id=...q=probability and measure billingsley&f=false
Example no 2.2 (section: Probability Measure), page 21.

Thanks in advance.
 
Last edited by a moderator:
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woundedtiger4 said:
A= {(0,0.1], (0.2, 0.3], (0.4, 0.5], (0.6, 0.7], (0.8, 1]}

What does this even mean? A is a set with intervals as elements?? That's not what Billingsley means.
 
Please ignore this post, as it is wrong

If you mean

A=(0,0.1]\cup (0.2, 0.3]\cup (0.4, 0.5]\cup (0.6, 0.7]\cup (0.8, 1]

then yes, Ac is the empty set, because A=(0,1].
 
Last edited:
micromass said:
If you mean

A=(0,0.1]\cup (0.2, 0.3]\cup (0.4, 0.5]\cup (0.6, 0.7]\cup (0.8, 1]

then yes, Ac is the empty set, because A=(0,1].

Sorry for the typing mistake, yes I mean union. but then why text says A^c = (0,a_1]U(a'_1, a_2]U...U(a'_m-1, a_m]U(a'_m, 1] , why does the complement have 0 & 1 in text ?

[STRIKE]Perhaps I am also wrong of being making the A^c = empty set because Ω contains only 0 & 1 so shouldn't it be A^c = {(0.1, 0.2], (0.3, 0.4], (0.5, 0.6], (0.7, 0.8]} ?[/STRIKE]
 
Last edited:
If A=(0,0.1]\cup (0.2, 0.3]\cup (0.4, 0.5]\cup (0.6, 0.7]\cup (0.8, 1], then the formula given states that A^{c} = (0, 0] \cup (.1, .2] \cup (.3, .4] \cup (.5, .6] \cup (.7, .8] \cup (1, 1], which is correct.
 
Citan Uzuki said:
If A=(0,0.1]\cup (0.2, 0.3]\cup (0.4, 0.5]\cup (0.6, 0.7]\cup (0.8, 1], then the formula given states that A^{c} = (0, 0] \cup (.1, .2] \cup (.3, .4] \cup (.5, .6] \cup (.7, .8] \cup (1, 1], which is correct.

OK
So,
If a1=0, then
A^c=(0,0]∪(a1′,a2]∪...∪(am′,1]=(a1′,a2]∪...∪(am′,1]
since (0,0]=∅. Likewise, if am′=1 we have
A^c=(0,a1]∪(a1′,a2]∪...∪(1,1]=(0,a1]∪...∪(am−1′,am]
since (1,1]=∅
 
micromass said:
If you mean

A=(0,0.1]\cup (0.2, 0.3]\cup (0.4, 0.5]\cup (0.6, 0.7]\cup (0.8, 1]

then yes, Ac is the empty set, because A=(0,1].
How comes? Shouldn't
Ac = {0} U (0.1, 0.2] U (0.3, 0.4] U (0.5, 0.6] U (0.7, 0.8] ?
 
woundedtiger4 said:
...since (0,0]=∅...

I think (0,0]= {0}.
 
xAxis said:
How comes? Shouldn't
Ac = {0} U (0.1, 0.2] U (0.3, 0.4] U (0.5, 0.6] U (0.7, 0.8] ?
No. (0,0] is the empty set. Other than that, you are correct, as is Citan Uzuki in post #5.
 

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