Sigma-Algebra is Delta-Algebra

Theorem
Every sigma-algebra is a delta-algebra.

Conversely, every delta-algebra is a sigma-algebra.

Proof
Let $$\mathcal S$$ be a sigma-algebra whose unit is $$\mathbb U$$.

Let $$A_1, A_2, \ldots$$ be a countably infinite collection of elements of $$\mathcal S$$.

Then:

$$ $$ $$ $$

And so $$\mathcal S$$ is also a delta-algebra.

Now let $$\mathcal D$$ be a delta-algebra whose unit is $$\mathbb U$$.

Let $$A_1, A_2, \ldots$$ be a countably infinite collection of elements of $$\mathcal D$$.

Then:

$$ $$ $$ $$

And so $$\mathcal D$$ is also a sigma-algebra.