Null Sets
A subset of Rn is a null set if, for every ε > 0, it can be covered with countably many products of n intervals whose total volume is at most ε. All countable sets are null sets.
If a subset of Rn has Hausdorff dimension less than n then it is a null set with respect to n-dimensional Lebesgue measure. Here Hausdorff dimension is relative to the Euclidean metric on Rn (or any metric Lipschitz equivalent to it). On the other hand a set may have topological dimension less than n and have positive n-dimensional Lebesgue measure. An example of this is the Smith–Volterra–Cantor set which has topological dimension 0 yet has positive 1-dimensional Lebesgue measure.
In order to show that a given set A is Lebesgue measurable, one usually tries to find a "nicer" set B which differs from A only by a null set (in the sense that the symmetric difference (A − B) (B − A) is a null set) and then show that B can be generated using countable unions and intersections from open or closed sets.
Read more about this topic: Lebesgue Measure
Famous quotes containing the words null and/or sets:
“A strong person makes the law and custom null before his own will.”
—Ralph Waldo Emerson (1803–1882)
“The world can doubtless never be well known by theory: practice is absolutely necessary; but surely it is of great use to a young man, before he sets out for that country, full of mazes, windings, and turnings, to have at least a general map of it, made by some experienced traveller.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)