# What is the intersection of a single set?

I'm currently reading through some set theory questions for a statistics class and I came across the following notation that I've never seen before, and I can't seem to find anywhere else online. The equation in question is: $$\mathcal{B} = \cap\{\mathcal{E}:\mathcal{D}\subset\mathcal{E} \text{ and } \mathcal{E} \text{ is a } \sigma \text{-field}\}$$ My question is, what does the intersection symbol in front of the set builder notation mean? How can I take the intersection of only one set? I would ask my professor, but they don't have any office hours today and I'd rather not wait too long for what seems like a simple question.

• Probably the intersection of all the valid $\mathcal E$. Commented Sep 28, 2020 at 18:57

The notation $$\cap A$$ usually denotes the intersection of all the sets that are contained in $$A$$, although not everyone uses this notation. Using the more common notion of an intersection over an index set, this means that
$$\cap A = \bigcap_{B\in A} B$$ holds, and that $$A$$ is a set of sets.
In your case, this means $$\mathcal{B} = \cap\{\mathcal{E}:\mathcal{D}\subset\mathcal{E} \text{ and } \mathcal{E} \text{ is a } \sigma \text{-field}\} = \bigcap_{\mathcal{E}:\mathcal{D}\subset\mathcal{E} \text{ and } \mathcal{E} \text{ is a } \sigma \text{-field}} \mathcal{E}$$
• So in this case, the result would be the smallest such $\mathcal{E}$? Commented Sep 28, 2020 at 19:07