Null set

As a null set (or null set) is called in mathematics a subset of a Maßraums (more precisely, is an element of the corresponding σ - algebra), which has dimension zero. It is not to be confused with the empty set; actually may contain infinitely many elements a zero quantity even. Some authors take in the definition of zero quantity should also negligible amounts, that is, those which subset of a null set, but not necessarily element of the algebra are and which therefore may itself not a measure is associated. If all amounts that differ only by such a negligible amount of an element of the algebra, also associated with a measure, it is called the completion of the measurement, as used for example in the definition of the Lebesgue measure.

From a property that applies to all elements of the Maßraums outside a measure zero, we say that it is true - almost everywhere. Is a probability measure, so we also say -almost surely instead of- almost everywhere.

Examples

• The empty set is in any measure space a null set.

For the Lebesgue measure on or following applies:

• Every countable subset of a null set. In particular, the set of rational numbers is a null set in the set of real numbers.
• Every genuine subspace, in particular any hyperplane of a set of measure zero. The same is true for affine subspaces and submanifolds whose dimension is less than.
• The Cantor set is an uncountable set of measure zero in the set of real numbers.

Generalization

One can also define more generally for elements of a half-ring null sets. A lot of is called null set if for the content. This generalization includes the above definition, since each algebra is also a half-ring and each measure is also a table of contents.