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.