In mathematics, a metric outer measure is an outer measure μ defined on the subsets of a given metric space (X, d) such that μ ( A ∪ B ) = μ ( A ) + μ...

dimension-like metric invariant now called Hausdorff dimension. Outer measures are commonly used in the field of geometric measure theory. Measures are generalizations...

sets is a measure. It is called the d {\displaystyle d} -dimensional Hausdorff measure of S {\displaystyle S} . Due to the metric outer measure property...

on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These conditions guarantee that the measure is "compatible" with...

regular measure. Any Borel probability measure on a locally compact Hausdorff space with a countable base for its topology, or compact metric space, or...

subset E ⊆ R {\displaystyle E\subseteq \mathbb {R} } , the Lebesgue outer measure λ ∗ ( E ) {\displaystyle \lambda ^{\!*\!}(E)} is defined as an infimum...

Borel measure μg on [a, b] which agrees with w on every interval I. The measure μg arises from an outer measure (in fact, a metric outer measure) given...

gauge corresponds to a smaller diameter needle. The French size is a measure of the outer diameter of a catheter (not internal drainage channel, or inner diameter)...

up in 1947.[citation needed] The "M" designation for metric screws indicates the nominal outer diameter of the screw thread, in millimetres. This is...

taking internal measures. The tape has three scales: the normal metric, the internal scale and a diameter scale used for instance to measure sheet metal to...

metric (QTM), but as one turn in the face metric (FTM, or HTM "Half Turn Metric", or OBTM "Outer Block Turn Metric"). The maximal number of face turns needed...

In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field...

via the Lebesgue measure. In the one-dimensional case, the Lebesgue outer measure of a set is defined in terms of the lengths of open intervals. Concretely...

advanced mathematics, the concept of distance has been generalized to abstract metric spaces, and other distances than Euclidean have been studied. In some applications...

supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit. The limit inferior of a sequence ( x n ) {\displaystyle (x_{n})}...

equivalent. If a Borel measure μ {\displaystyle \mu } is both inner regular and outer regular, it is called a regular Borel measure. If μ {\displaystyle...

of the metric system and the world's most widely used system of measurement. Coordinated by the International Bureau of Weights and Measures (abbreviated...

properly metrized by the induced metric. If the measure space is separable, it can be shown that the corresponding metric space is, too. Let X {\displaystyle...

Lebesgue measure Lorentz space Lifting theory Measurable cardinal Measurable function Minkowski content Outer measure Product measure Pushforward measure Regular...

the metric WGS-84 datum, which is also used by the Global Positioning System. An ISO 2848 measure of 3 basic modules (30 cm) is called a "metric foot"...

(ISO-Metric) standards specify the load index rating at an inflation pressure of 36 psi (250 kPa) (table below), while P-Metric standards measure the load...

designed so as not to measure randomness as complex and has been useful in many applications. More recently, a complexity metric was developed for images...

Hausdorff measure, a fractional-dimension analogue of the Lebesgue measure. First, an outer measure is constructed: Let X {\displaystyle X} be a metric space...

alternatively, 0.0001 inch = 0.00254 millimeters. Therefore, metric micrometers provide smaller measuring increments than comparable inch unit micrometers—the...

metric (FLRW; /ˈfriːdmən ləˈmɛtrə ... /) is a metric based on an exact solution of the Einstein field equations of general relativity. The metric describes...

of the French Revolution was the dramatic rationalization of measures as the new metric system. The change was extremely unpopular, however, and a metricized...

has the same 60° profile as the ISO metric screw thread, but the characteristic dimensions of each UTS thread (outer diameter and pitch) were chosen as...

Y)}{\mathrm {H} (X,Y)}}\leq 1.} The metric D {\displaystyle D} is a universal metric, in that if any other distance measure places X {\displaystyle X} and...

fusion of hydrogen into helium at its core, releasing this energy from its outer photosphere. Astronomers classify it as a G-type main-sequence star. The...

^{2}}}}{2}}} where the purely radial component grr of the metric goes to infinity. The outer surface can be approximated by an oblate spheroid with lower...