Extending a set family into a sigma algebra

Expert intro: This question asks you to inspect a family of subsets on an interval, verify sigma algebra axioms, and then enlarge the family just enough to satisfy closure without becoming the power set.

Detailed walkthrough

Start with the sigma algebra conditions

A $\sigma$-algebra on a set $X$ must contain $\varnothing$ and $X$, and it must be closed under complements and countable unions. Here the universe is the interval $X=[0,2]$.

The given family includes singletons like $\{0\}$, $\{1\}$, and $\{2\}$, as well as intervals such as $(0,2]$ and $[0,1)\cup(1,2]$. To decide whether it is already a sigma algebra, you must check complements relative to $[0,2]$.

Check the complements

Take $\{0\}$. Its complement in $[0,2]$ is $(0,2]$, which is listed. Take $\{1\}$. Its complement in $[0,2]$ is $[0,1)\cup(1,2]$, which is also listed. Take $\{2\}$. Its complement is $[0,2)$, but that set is not in the collection.

Because one complement is missing, the family is not yet a sigma algebra.

How to extend it without using the power set

To extend the family, you add the missing complements and then close under unions. The smallest sigma algebra containing the given sets is generated by the atoms created by the points $0$, $1$, and $2$ inside $[0,2]$. That means you must include all sets formed by unions of the basic pieces determined by those points.

A proper extension would add the missing interval pieces needed to make every listed set have its complement present, while still stopping short of the full power set of $[0,2]$. The final family should be closed under complement and union, but not contain arbitrary subsets of the continuum.

Main insight

For set families on an interval, singletons can force the inclusion of many related pieces. The safest method is to identify the partition induced by the listed sets, then take all unions of the resulting atoms. That gives the generated sigma algebra and avoids overextending to the entire power set.

💡 Pitfall guide

A typical mistake is to think that because the collection contains $\varnothing$, $X$, some singletons, and some interval unions, it must already be a sigma algebra. It does not: every listed set needs its complement in the same family. Another error is extending by adding random missing sets without checking closure under unions, which can produce a family that is still incomplete. When the universe is an interval, be careful with endpoints. The complement of $\{2\}$ inside $[0,2]$ is not the same as the complement in $\mathbb R$, and that distinction matters here.

🔄 Real-world variant

If the family were changed to $\{\varnothing,\{0\},[0,2) ,X\}$, then the sigma algebra generated by it would be different: the point $0$ creates one atom and the interval $[0,2)$ creates another, so the generated family would be much smaller than the full power set. If instead the family included all three singletons $\{0\}$, $\{1\}$, and $\{2\}$ together with the necessary complements, the generated sigma algebra would be the one determined by the partition $\{\{0\},\{1\},\{2\},(0,1), (1,2)\}$, leading to all unions of those atoms. The exact extension always depends on which basic distinctions the original sets force.

🔍 Related terms

generated sigma algebra, set complement, atoms of a partition