Definition [efr-YGH6]

Let (X, \Sigma ) be a measurable space. A \sigma -ideal is a subset N \subset \Sigma which is downwards stable (A \subset B, B \in N implies A \in N) and stable under countable unions.

A measurable space equipped with a \sigma -ideal (X,\Sigma ,N) is called an enhanced measurable space. Note that if (X,\Sigma ,\mu ) is a measure space, the collection of nullsets turns X into an enhanced measurable space. If there is no chance of confusion, we will use this implicitly.

We also use the terminology from measure spaces for enchanced measurable spaces---thus we refer to the sets in N as nullsets and say that two functions which differ only on a nullset are equal almost everywhere, and so on.