Relation of ideals in probability with other kinds of ideals?

Question

It seems that there are at least 5 kinds of ideals in maths:

1. Ideals in number theory (Kummer, Dedekind)
2. Ideals in abstract algebra (Dedekind, Noether), as kernels of homomorphisms
3. Ideals in order theory (Marshall H. Stone), as order-preserving lattices
4. Ideals in set theory (wnich author could be quoted here as the initiator?): There seems to be a direct translation to 3.
5. Ideals in probability.

I would like to know:

In which way the concept in probability is defined and how it is related to the other meaninfs of the term. Who could be the reference author in that case? An author for 4) would be also welcome.

Thanks in advance.