It can be proven by using Fatou's lemma and the properties of null sets.

Sets of measure 0, called null sets, are negligible.

In the null set, this is always false.

A subset of a null set is called a negligible set.

A standard probability space may contain a null set of any cardinality, thus, it need not be countably separated.

The empty set is always a null set.

More generally, any countable union of null sets is null.

A measure in which all subsets of null sets are measurable is complete.

Then the following inequality is true where is the null set.

The truth of these last statements, when used in a free logic, depend on the domain of quantification, which may be the null set.