Null Set

A null set in measure theory is a measurable set whose measure is zero. For Lebesgue measure, this means zero length on the real line, zero area in the plane, or zero volume in higher dimensions. Nullity depends on the measure being used: a set can be null for one measure but not for another. Unlike the empty set, a null set may contain infinitely many points. The concept underlies the treatment of exceptional sets in integration and probability. (math.ucdavis.edu)

Definition and basic properties

Let $(X,\Sigma,\mu)$ be a measure space, where $\Sigma$ is a sigma-algebra of measurable subsets of $X$. A set $N\in\Sigma$ is null with respect to $\mu$ if

The empty set is always null. By monotonicity, every measurable subset of a null set is also null. Furthermore, a countable union of measurable null sets is null, since countable subadditivity gives

No disjointness assumption is needed. (math.ucdavis.edu)

The restriction to countable unions is essential. Every singleton in the real line has Lebesgue measure zero, but their uncountable union is the entire line. In other measure spaces, even singletons need not be null: under counting measure, each singleton has measure one, so only the empty set is null. Consequently, “small” here means negligible for the chosen measure, not necessarily small in cardinality. (math.ucdavis.edu)

Lebesgue-null sets and coverings

For a subset $N$ of Euclidean space $\mathbb R^n$, Lebesgue nullity has a geometric characterization. For every $\varepsilon>0$, there must be a countable collection of rectangles $R_k$ such that

Thus the set can be covered with arbitrarily small total volume. This does not require the set itself to be bounded or to have a simple shape. The criterion says its Lebesgue outer measure is zero; every such set is Lebesgue measurable. (math.ucdavis.edu)

Every countable set of real numbers is null. For an enumeration $N=\{x_1,x_2,\ldots\}$, cover $x_k$ by an interval of length $\varepsilon 2^{-k-1}$. The total length is less than $\varepsilon$. In particular, the rational numbers form a null set, even though they are a dense set in the real line. Density therefore does not imply positive measure. (math.ucdavis.edu)

The standard middle-thirds Cantor set is an uncountable null set. After $k$ construction stages, it is contained in $2^k$ intervals, each of length $3^{-k}$. Their total length is $(2/3)^k$, which tends to zero. This example separates measure zero from countability. Likewise, a line segment in the plane has zero two-dimensional Lebesgue measure, despite having positive length when measured along the segment. (maths-people.anu.edu.au)

Completeness and measurability

A measure space is complete if every subset of every measurable null set is measurable. In a noncomplete space, a subset of a null set may lie outside the sigma-algebra; its measure is then undefined, rather than automatically zero. Some authors use “null” more broadly for subsets contained in measurable zero-measure sets, so the convention matters. (math.ucdavis.edu)

Lebesgue measure on the Borel sigma-algebra is not complete: the Cantor set has non-Borel subsets, although it is itself Borel and null. Completing this measure adds all subsets of Borel null sets and produces the Lebesgue sigma-algebra. This explains why Borel measurability and Lebesgue measurability are distinct. (math.ucdavis.edu)

Almost-everywhere equality and integration

A property holds almost everywhere if its exceptions are confined to a null set. Two measurable functions $f$ and $g$ are equal almost everywhere when they agree outside such a set. For Lebesgue measure, changing a measurable function arbitrarily on a null set preserves measurability. More generally, this assertion requires completeness of the measure space. (ocw.mit.edu)

Null sets do not contribute to a Lebesgue integral. For measurable $N$,

where $\mathbf 1_N$ is its indicator function. Measurable functions equal almost everywhere therefore have the same integral whenever it is defined. In Lp spaces, functions are identified through equivalence classes under almost-everywhere equality, rather than treated as distinct merely because their pointwise values differ on null sets. (ocw.mit.edu)

Null events in probability

In a probability space $(\Omega,\mathcal F,P)$, a measurable event $A$ is null if $P(A)=0$. A property holds almost surely when its exceptional event is null. This is the probabilistic counterpart of almost-everywhere validity. (statlect.com)

Zero probability does not imply that an event is empty. For a random variable $U$ with the uniform distribution on $[0,1]$, each event $\{U=a\}$, with $a\in[0,1]$, has probability zero, although these events together cover all possible outcomes. Probability measures are countably additive, not arbitrarily additive; one cannot sum probabilities over uncountably many singleton events as though the usual countable-additivity rule applied. (statlect.com)