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 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 ).

If X is Cauchy distributed with median μ and scale parameter γ, then the complex variable

If there is a change in the potential energy of a system ; for example μ < sub > 1 </ sub >> μ < sub > 2 </ sub > ( μ is Chemical potential ) an energy flow will occur from S < sub > 1 </ sub > to S < sub > 2 </ sub >, because nature always prefers low energy and maximum entropy.

If μ is positive, the origin will be shifted to the right, and if μ is negative, it will be shifted to the left.

If μ is an unknown constant and W is random noise with probability density, then has probability density and is therefore a location family ..

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 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

If μ is a positive measure, then N is null ( or zero measure ) if its measure μ ( N ) is zero.

If μ is not a positive measure, then N is μ-null if N is | μ |- null, where | μ | is the total variation of μ ; equivalently, if every measurable subset A of N satisfies μ ( A )

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.

If the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

If you take the one, you'd better take both ''.

If we was both armed, you wouldn't talk so tough ''.

If life and death did not both present themselves to us, there would be no inscrutability.

If a broader Atlantic community is to be formed -- and my own judgment is that it lies within the realm of both our needs and our capacity -- a ready nucleus of machinery is at hand in the NATO alliance.

If necessary to replace both halves on grill, sear cuts and allot extra time.

If that's done, the house can be designed and oriented for best operation, and this can mean savings both in the size of equipment and in the cost of the house itself.

If both George and a piece of information George does not have are prominent in the context, but the idea of telling George is new, then dominant stress will probably be on tell in why not tell George??

If the word deliberate means anything, both of them certainly deliberately destroyed themselves.

If these allegiances come into conflict, he or she may be guilty of treason against one or both.

If two players tie for majority, they will share both shareholder bonuses.

If instead Betty and Carla both owned 7 shares, then Alex would keep the majority bonus for herself, while Betty and Carla would split the minority bonus.

If both alleles are the same, they are homozygotes.

If through lack of issue, marriage or both, eventually only one person represents the claims of all the sisters, he or she can claim the dignity as a matter of right, and the abeyance is said to be terminated.

If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

If each player wins a set, or both sets end tied, there is a 3-end tiebreaker to determine a winner.

If either one die or the other but not both can be moved, the higher must be used.

" If both opponents roll the same opening number, the doubling cube is incremented on each occasion yet remains in the middle of the board, available to either player.

If both extra periods are played without a scored goal, a penalty shootout will settle the game.

If the same state vector appears on both bra and ket side,

If we allow both, the problem is called the quantified Boolean formula problem ( QBF ), which can be shown to be PSPACE-complete.

If the page's creator and reader are both assuming some platform-specific character encoding, and the server does not send any identifying information, then the reader will nonetheless see the page as the creator intended, but other readers on different platforms or with different native languages will not see the page as intended.

If the sets are short to medium length the caller will often try to run the dance until each couple has danced with every other couple both as a 1 and a 2 and returned to where they started.