The symbol μ denotes a positive measure: that is, a real-valued positive set function defined on a σ-algebra which is countably additive.

Any measure μ defined on the σ-algebra of Borel sets is called a Borel measure.

Some authors require in addition that μ ( C ) < ∞ for every compact set C. If a Borel measure μ is both inner regular and outer regular, it is called a regular Borel measure.

If μ is both inner regular and locally finite, it is called a Radon measure.

Note that a locally finite Borel measure automatically satisfies μ ( C ) < ∞ for every compact set C.

In this case, is the smallest σ-algebra that contains the open intervals of R. While there are many Borel measures μ, the choice of Borel measure which assigns for every interval is sometimes called " the " Borel measure on R. In practice, even " the " Borel measure is not the most useful measure defined on the σ-algebra of Borel sets ; indeed, the Lebesgue measure is an extension of " the " Borel measure which possesses the crucial property that it is a complete measure ( unlike the Borel measure ).

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

μ is a finite measure on the sigma-algebra, so that the triplet ( X, Σ, μ ) is a probability space.

A measure μ is monotonic: If E < sub > 1 </ sub > and E < sub > 2 </ sub > are measurable sets with E < sub > 1 </ sub > ⊆ E < sub > 2 </ sub > then

A measure μ is continuous from below: If E < sub > 1 </ sub >, E < sub > 2 </ sub >, E < sub > 3 </ sub >, … are measurable sets and E < sub > n </ sub > is a subset of E < sub > n + 1 </ sub > for all n, then the union of the sets E < sub > n </ sub > is measurable, and

A measure μ is continuous from above: If E < sub > 1 </ sub >, E < sub > 2 </ sub >, E < sub > 3 </ sub >, … are measurable sets and E < sub > n + 1 </ sub > is a subset of E < sub > n </ sub > for all n, then the intersection of the sets E < sub > n </ sub > is measurable ; furthermore, if at least one of the E < sub > n </ sub > has finite measure, then

There was a measure of protection in its concrete walls and ceiling, but the engineers who hastily installed it were well aware that concrete is not much better than prayer, if as efficacious, when a direct hit comes along.

But the problem is one which gives us the measure of a man, rather than a group of men, whether a group of doctors, a group of party members assembled at a dinner to give their opinion, or the masses of the voters.

The measure of combat efficiency in an indecisive campaign is a matter of personal choice.

but this -- yes, terrible step I am about to take is lightened with an inundating joy by the new-found hope that here, in these poems, is treasure -- or at least some measure of beauty, which I did not know of ''.

Such a list must naturally be selective, and the treatment of each man is brief, for I am interested only in their general ideas on the moral measure of literature.

When we turn to Aristotle's ideas on the moral measure of literature, it is at once apparent that he is at times equally concerned about the influence of the art.

But if any realism and feeling for truth remain in the General Assembly, it is time for men of courage to measure the magnitude of the failure and urge some new approach.

The development and testing of new apparatus to measure other properties is nearing completion.

In the case of taxpaying corporate stockholders, the measure would be the lesser of the fair market value of the shares or Du Pont's tax basis for them, which is approximately $2.09 per share.

To measure the volume of one of the combustion chambers in the cylinder head, install the valves and spark plug in the chamber and support the head so that its gasket surface is level.

No attempts to measure the radio emission of the remaining planets have been reported, and, because of their distances, small diameters, or low temperatures, the thermal radiation at radio wave lengths reaching the earth from these sources is expected to be of very low intensity.

Such an instrument is expected to be especially useful if it could be used to measure the elasticity of heavy pastes such as printing inks, paints, adhesives, molten plastics, and bread dough, for the elasticity is related to those various properties termed `` length '', `` shortness '', `` spinnability '', etc., which are usually judged by subjective methods at present.

A measure of the total mass accretion of meteoritic material by the Earth is obtained from analyses of deep-sea sediments and dust collected in remote regions ( Pettersson, 1960 ).

Acceptance of radiopasteurization is likely to be delayed, however, for two reasons: ( 1 ) the storage life of fresh chicken under refrigeration is becoming a minimal problem because of constantly improved sanitation and distributing practices, and ( 2 ) treatment by antibiotics, a measure already approved by the Federal Food and Drug Administration, serves to extend the storage life of chicken at a low cost of about 0.5 cents per pound.

Since the removal force is a function of coating thickness, a differential transformer pickup has been incorporated into the instrument to accurately measure film thickness.

It is difficult to measure the direction and magnitude of R directly.

A reflectance-measuring instrument may be desirable to measure cleaning, whereas Soxhlet extraction is necessary to measure grease removal.

Generally, it is necessary to mark distances on a specimen ( or garment ) in both lengthwise and widthwise directions and to measure before and after laundering.

The objective function is some measure of the increase in value of the stream by processing ; ;

Part-time farming gives a measure of security if the regular job is lost, provided the farm is owned free of debt and furnishes enough income to meet fixed expenses and minimum living costs.

Now it is easy to convince oneself that the set X could not possibly be measurable for any rotation-invariant countably additive finite measure on S. Hence one couldn't expect to find an algorithm to find a point in each orbit, without using the axiom of choice.

It must assign 0 to the empty set and be ( countably ) additive: the measure of a " large " subset that can be decomposed into a finite ( or countable ) number of " smaller " disjoint subsets, is the sum of the measures of the " smaller " subsets.

Several further properties can be derived from the definition of a countably additive measure.

Consider the closed intervals for all integers k ; there are countably many such intervals, each has measure 1, and their union is the entire real line.

In other words, the probability is obtained by integrating the characteristic function of B against the countably additive measure

A non-negative countably additive Borel measure μ on a locally compact Hausdorff space X is regular if and only if

If μ is a complex-valued countably additive Borel measure, μ is regular iff the non-negative countably additive measure | μ | is regular as defined above.

For any continuous linear functional ψ on C < sub > 0 </ sub >( X ), there is a unique regular countably additive complex Borel measure μ on X such that

There is, up to a positive multiplicative constant, a unique countably additive, nontrivial measure μ on the Borel subsets of G satisfying the following properties:

The probability measure function must satisfy a simple requirement: the probability of a union of two ( or countably many ) disjoint events must be equal to the sum of probabilities of each of these events.

In measure theory, if we have a countably infinite collection of measurable sets, then the union and intersection of all of them is a measurable set.

The measure-theoretic formulation of Kolmogorov assumes that a probability measure is countably additive.

A multiplication operator is defined as follows: Let be a countably additive measure space and f a real-valued measurable function on X.

A real valued measurable cardinal less than or equal to exists if there is a countably additive extension of the Lebesgue measure to all sets of real numbers.

More precisely, a right Haar measure on a locally compact group G is a countably additive measure μ defined on the Borel sets of G which is right invariant in the sense that μ ( A x ) = μ ( A ) for x an element of G and A a Borel subset of G and also satisfies some regularity conditions ( spelled out in detail in the article on Haar measure ).

For the purposes of functional analysis, and in particular of harmonic analysis, one wishes to carry over the group ring construction to topological groups G. In case G is a locally compact Hausdorff group, G carries an essentially unique left-invariant countably additive Borel measure μ called Haar measure.

* sigma-algebras, sigma-fields and sigma-finiteness in measure theory ; more generally, the symbol σ serves as a shorthand for " countably ", e. g. a σ-compact topological space is one that can be written as a countable union of compact subsets.