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By definition, any Radon measure is locally finite. The counting measure is sometimes locally finite and sometimes not: the counting measure on the integers with their usual discrete topology is locally finite, but the counting measure on the real line with its usual Borel topology is not.
In mathematics, a measure is said to be saturated if every locally measurable set is also measurable. [1] A set E {\displaystyle E} , not necessarily measurable, is said to be a locally measurable set if for every measurable set A {\displaystyle A} of finite measure, E ∩ A {\displaystyle E\cap A} is measurable.
[1] [2] Carathéodory's work on outer measures found many applications in measure-theoretic set theory (outer measures are for example used in the proof of the fundamental Carathéodory's extension theorem), and was used in an essential way by Hausdorff to define a dimension-like metric invariant now called Hausdorff dimension.
It turns out that pre-measures give rise quite naturally to outer measures, which are defined for all subsets of the space . More precisely, if is a pre-measure defined on a ring of subsets of the space , then the set function defined by = {= |, =} is an outer measure on and the measure induced by on the -algebra of Carathéodory-measurable sets satisfies () = for (in particular, includes ).
In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing. Outerplanar graphs may be characterized (analogously to Wagner's theorem for planar graphs) by the two forbidden minors K 4 and K 2,3, or by their Colin de Verdière graph invariants. They have Hamiltonian ...
An example of a measure on the real line with its usual topology that is not outer regular is the measure where () =, ({}) =, and () = for any other set .; The Borel measure on the plane that assigns to any Borel set the sum of the (1-dimensional) measures of its horizontal sections is inner regular but not outer regular, as every non-empty open set has infinite measure.
Also confidence coefficient. A number indicating the probability that the confidence interval (range) captures the true population mean. For example, a confidence interval with a 95% confidence level has a 95% chance of capturing the population mean. Technically, this means that, if the experiment were repeated many times, 95% of the CIs computed at this level would contain the true population ...
An informative prior expresses specific, definite information about a variable. An example is a prior distribution for the temperature at noon tomorrow. A reasonable approach is to make the prior a normal distribution with expected value equal to today's noontime temperature, with variance equal to the day-to-day variance of atmospheric temperature, or a distribution of the temperature for ...