# Sigma algebras on [0,1]

1. Dec 7, 2012

### Old Monk

This topic came up while studying measures on sub-intervals of [0,1]. The collection of all intervals in [0,1] is a semi-algebra, say J. Now from finite disjoint union of members of J lets say we form a set A.

I was able to prove that A is an algebra, since for any C,D ε A, C$\cap$D and C$^{c}$ belong to A.

I'm not able to understand why A isn't a σ-algebra. Can anyone please outline a proof or give me a counter-argument.

2. Dec 7, 2012

### micromass

Staff Emeritus
Are singletons in A?? Is $\mathbb{Q}$ in A?

3. Dec 7, 2012

### Old Monk

Yeah singletons are in A. Since singletons of the form (a,a) exist in J, the single union of such elements should exist in A. That I assume implies, all the rationals in [0,1] are also in A.

4. Dec 7, 2012

### micromass

Staff Emeritus
Isn't $(a,a)=\emptyset$?? Singletons would rather come from $[a,a]=\{a\}$. But ok, let's assume that the singletons are in A.

Why??

5. Dec 7, 2012

### Old Monk

Yeah sorry, that's [a,a].

We have [0,r)$\cup$(r,1] in A, if r is any rational in [0,1]. Therefore the element {r} ε A. So I think we could generalize this to all r ε Q$\cap$[0,1].

6. Dec 7, 2012

### micromass

Staff Emeritus
Yeah, I agree that every {r} is in A. But why does that imply that $\mathbb{Q}\in A$. Is it even true??