## 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 .

[...] There are a bunch of useful facts that we can prove with the help of measurable covers and measurable kernels. These will allow us to move statements we want to show about outer and inner measures to the realm [...]

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[...] Using Measurable Covers and Kernels II Following yesterday’s post, here are some more useful facts that we can prove with the help of measurable covers and measurable kernels. [...]

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