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 countably subadditive: If E < sub > 1 </ sub >, E < sub > 2 </ sub >, E < sub > 3 </ sub >, … is a countable sequence of sets in Σ, not necessarily disjoint, then

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.

The goal of DSP is usually to measure, filter and / or compress continuous real-world analog signals.

An analog joystick is a joystick which has continuous states, i. e. returns an angle measure of the movement in any direction in the plane or the space ( usually using potentiometers ), whereas a digital joystick gives only on / off signals for four different directions and its mechanically possible combinations ( such as up-right or down-left ).

The more mathematically advanced measure theory based treatment of probability covers both the discrete, the continuous, any mix of these two and more.

Since there is a one-to-one correspondence between Borel regular measures in the interval and functions of bounded variation ( that assigns to each function of bounded variation the corresponding Lebesgue-Stieltjes measure, and the integral with respect to the Lebesgue-Stieltjes measure agrees with the Riemann-Stieltjes integral for continuous functions ), the above stated theorem generalizes the original statement of F. Riesz.

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

The measure δ is not absolutely continuous with respect to the Lebesgue measure — in fact, it is a singular measure.

Only random vectors whose distributions are absolutely continuous with respect to a measure are said to have densities ( with respect to that measure ).

* A Lipschitz function g: R → R is absolutely continuous and therefore is differentiable almost everywhere, that is, differentiable at every point outside a set of Lebesgue measure zero.

More specifically, the energy Eigen states form a basis-any wavefunction may be written as a sum over the discrete energy states or an integral over continuous energy states, or more generally as an integral over a measure.

The Wiener measure is the probability law on the space of continuous functions g, with g ( 0 ) = 0, induced by the Wiener process.

There are, however, practical measurement problems no matter which definition is used, because it is difficult to measure continuous exposure to the risk of pregnancy over a period of years.

The gesture leading up to the ictus is called the " preparation ", and the continuous flow of steady beats is called the "" ( the German word for bar or measure ).

Both types provide a continuous record of ground motion ; this distinguishes them from seismoscopes, which merely indicate that motion has occurred, perhaps with some simple measure of how large it was.

In 1967 the Czech physiologist Jan Peñáz invented and patented the volume clamp method to measure continuous blood pressure.

In the continuous setting, a weight is a positive measure such as ' on some domain, which is typically a subset of a Euclidean space, for instance could be an interval.

A meaningful and reliable measure of the maximum power output of an audio amplifier – or the power handling of a loudspeaker – is continuous average sine wave power.

Connectivity is the measure of how connected or spatially continuous a corridor, network, or matrix is.

The Cantor function challenges naive intuitions about continuity and measure ; though it is continuous everywhere and has zero derivative almost everywhere, c goes from 0 to 1 as x goes from 0 to 1, and takes on every value in between.