null set

Then the following inequality is true where is the null set.

In the null set, this is always false.

A subset of a null set is called a negligible set.

A measure in which all subsets of null sets are measurable is complete.

Null set, a set that is negligible in some sense

The empty set is always a null set.

More generally, any countable union of null sets is null.

It can be proven by using Fatou's lemma and the properties of null sets.

Almost everywhere, a mathematical expression for a property holding on the complement of a null set

Sets of measure 0, called null sets, are negligible.

Null set was once a common synonym for "empty set", but is now a technical term in measure theory.

The ideal of null sets in measure theory

Thus, null sets may be interpreted as negligible sets, defining a notion of almost everywhere.

A standard probability space may contain a null set of any cardinality, thus, it need not be countably separated.

Mikusinski's lemma allows one to define integral without mentioning null sets.

The truth of these last statements, when used in a free logic, depend on the domain of quantification, which may be the null set.

A measure space is complete if every subset of every null set is measurable.

More generally, whenever an ideal is taken as understood, then a null set is any element of that ideal.

In measure theory, any set of measure 0 is called a null set (or simply a measure-zero set).

In particular, the set Q of rational numbers is a null set, despite being dense in R.

Or, let X be a measure space, and let negligible sets be the null sets.

All countable sets are null sets.

This invariant is called the uniformity of the ideal of null sets, denoted .

re a null set, dead or alive ?

More generally, any event (not necessarily in ) happens almost surely if is contained in a null set: a subset of some such that .