## Measurable Kernels

A measurable kernel is the flip side of a measurable cover. Specifically, given , a measurable kernel of is a set such that , and if for every with we have . And, as it happens, every set has a measurable kernel.

To find it, let be a measurable cover of . Then let be a measurable cover of , and set . Since contains , we find

.

If , then

Since was picked to be a measurable cover of , we conclude that , as we hoped.

Now if is a measurable kernel of , then . Indeed, since , we have . If this inequality is strict then , and there must be some with and . But , while , contradicting the fact that was chosen to be a measurable kernel of .

The symmetric difference of any two measurable kernels is negligible. Given two measurable kernels and , we know that . This implies that , and thus . Similarly, , and thus .