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Monoids, groups, and rings must have at least one element, while division algebras and fields must have at least two elements.
In mathematical sets, the null set, also called the empty set, is the set that does not contain anything. The null set makes it possible to explicitly define the results of operations on certain sets that would otherwise not be explicitly definable.
Since g(x) is strictly monotonic and continuous, it is a homeomorphism. However, if it were Borel measurable, then g(F) would also be Borel measurable (here we use the fact that the preimage of a Borel set by a continuous function is measurable; g(F) = (g.) Therefore, F is a null, but non-Borel measurable set.
In a separable Banach space (X, ), the group operation moves any subset A ⊂ X to the translates A x for any x ∈ X.
We need a strictly monotonic function, so consider g(x) = f(x) x. Because g is injective, we have that F ⊂ K, and so F is a null set.
Any measurable subset of a null set is itself a null set.
Together, these facts show that the m-null sets of X form a sigma-ideal on X.
This is because there is logically only one way that a set can contain nothing.
The intersection of two disjoint sets (two sets that contain no elements in common) is the null set.
When there is a probability measure μ on the σ-algebra of Borel subsets of X, such that for all x, μ(A x) = 0, then A is a Haar null set.