# Measure theory Question - Help

1. May 17, 2012

### woundedtiger4

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 [Broken]
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 lets 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.

Last edited by a moderator: May 6, 2017
2. May 17, 2012

### micromass

Staff Emeritus
What does this even mean??? A is a set with intervals as elements?? That's not what Billingsley means.

3. May 17, 2012

### micromass

Staff Emeritus
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: May 24, 2012
4. May 17, 2012

### woundedtiger4

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: May 17, 2012
5. May 17, 2012

### Citan Uzuki

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.

6. May 18, 2012

### woundedtiger4

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]=∅

7. May 24, 2012

### xAxis

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] ?

8. May 24, 2012

### xAxis

I think (0,0]= {0}.

9. May 24, 2012

### D H

Staff Emeritus
No. (0,0] is the empty set. Other than that, you are correct, as is Citan Uzuki in post #5.