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 A is an open or closed subset of R < sup > n </ sup > ( or even Borel set, see metric space ), then A is Lebesgue measurable.

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 ( X, Σ ) and ( Y, Τ ) are Borel spaces, a measurable function f: ( X, Σ ) → ( Y, Τ ) is also called a Borel function.

If ( X, Σ ) is some measurable space and A ⊂ X is a non-measurable set, i. e. if A ∉ Σ, then the indicator function 1 < sub > A </ sub >: ( X, Σ ) → R is non-measurable ( where R is equipped with the Borel algebra as usual ), since the preimage of the measurable set

Given the representation of T as a multiplication operator, it is easy to characterize the Borel functional calculus: If h is a bounded real-valued Borel function on R, then h ( T ) is the operator of multiplication by the composition.

If H is Hermitian ( the name for self-adjoint in the physics literature ) and f is a Borel function,

* If A is a Borel set, so is.

* If A < sub > n </ sub > is a Borel set for each natural number n, then the union is a Borel set.

If X is a topological space, and Σ is the sigma-algebra of Borel sets in X, then is the subspace of consisting of all regular Borel measures on X.

If E is a Borel subset of R, and 1 < sub > E </ sub > is the indicator function of E, then 1 < sub > E </ sub >( T ) is a self-adjoint projection on H. Then mapping

If y is a power series that converges in some neighborhood of the origin then it has a Borel sum at some point z if it can be analytically continued to a disc with diameter 0z.

If is a second-countable Hausdorff space and contains the Borel sigma-algebra, then the set of recurrent points of has full measure.

If Af is the change per unit volume in Gibbs function caused by the shear field at constant P and T, and **yr is the density of the fluid, then the total potential energy of the system above the reference height is Af.

If a union cannot perform this function, then collective bargaining is being palmed off by organizers as a gigantic fraud.

If the Greek letter is used, it is assumed to be a Fourier transform of another function,

If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of X.

If the function R is well-defined, its value must lie in the range, with 1 indicating perfect correlation and − 1 indicating perfect anti-correlation.

If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.

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 ).

If f is not a function, but is instead a partial function, it is called a partial operation.

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.

If your side has two aces and a void, then you are not at risk of losing the first two tricks, so long as ( a ) your void is useful ( i. e., does not duplicate the function of an ace that your side holds ) and ( b ) you are not vulnerable to the loss of the first two tricks in the fourth suit ( because, for instance, one of the partnership hands holds a singleton in that suit or the protected king, giving your side second round control ).

If evolutionary processes are blind to the difference between function F being performed by conscious organism O and non-conscious organism O *, it is unclear what adaptive advantage consciousness could provide.

If the wave function is regarded as ontologically real, and collapse is entirely rejected, a many worlds theory results.

If wave function collapse is regarded as ontologically real as well, an objective collapse theory is obtained.

If we take the simple valence bond structure and mix in all possible covalent and ionic structures arising from a particular set of atomic orbitals, we reach what is called the full configuration interaction wave function.

If we take the simple molecular orbital description of the ground state and combine that function with the functions describing all possible excited states using unoccupied orbitals arising from the same set of atomic orbitals, we also reach the full configuration interaction wavefunction.

If this happens, there may be some class conflict in the South, with school boards and school teachers taking the middle-class position.

If that happens, then he will be holding worthless stock at the end of the game.

If the relative priorities of these substituents need to be established, R takes priority over S. When this happens, the descriptor of the stereocenter is a lowercase letter ( r or s ) instead of the uppercase letter normally used.

If this process happens faster than it is naturally quenched by ions recombining, the new ions multiply in successive cycles until the gas breaks down into a plasma and current flows freely in a discharge.

If this happens to all industries however, everyone would be worse off than if they had been subject to the rigours of market competition.

If this happens rapidly before significant decay to the organic tissue, very fine three-dimensional morphological detail can be preserved.

If a statement happens to be true in our world, but is false in some other worlds, then it is a contingent truth.

# If the borrowed word happens to have a suffix that the borrowing language uses as a gender marker, the suffix tends to dictate gender.

If one thinks in this manner, then consider what happens when the story is extended further.

If the manoeuver is timed properly, Mars will be " arriving " under the spacecraft when this happens.

To Axel Oxenstierna, he confessed: " If anything happens to me, my family will merit your pity, the mother lacking in common sense, the daughter a minor-hopeless, if they rule, and dangerous, if others come to rule over them.

If no additional energy is supplied by any other mechanism, the nucleus will not fission, but will merely absorb the neutron, as happens when U-238 absorbs slow and even some fraction of fast neutrons, to become U-239.

The Code of Canon Law 332 § 2 states, " If it happens that the Roman Pontiff resigns his office, it is required for validity that the resignation is made freely and properly manifested but not that it is accepted by anyone.

If this happens, normal drainage of mucus within the sinuses is disrupted, and sinusitis may occur.

If most evolution happens in these rare instances of allopatric speciation then evidence of gradual evolution in the fossil record should be rare.

If this happens, the resulting lever angle created will not be as effective as if aligned with the oarlock or slightly behind it.

If the payout is 4, 000 times the input amount, and it happens every 4, 000 times on average, the return to player is exactly 100 %, but the game would be dull to play.

If this happens in an even dimension, the Majorana spinor representation will sometimes decompose into two Majorana – Weyl spinor representations.

If the demand decreases, then the opposite happens: a shift of the curve to the left.

If the quantity supplied decreases, the opposite happens.

If the third offence happens, 2 points will be given to the player.

If that happens, Franks said, “... the Western world, the free world, loses what it cherishes most, and that is freedom and liberty we ’ ve seen for a couple of hundred years in this grand experiment that we call democracy .” Franks then offered “ in a practical sense ” what he thinks would happen in the aftermath of such an attack.

If emergence happens over disparate size scales, then the reason is usually a causal relation across different scales.

If the convex hull of X is a closed set ( as happens, for instance, if X is a finite set or more generally a compact set ), then it is the intersection of all closed half-spaces containing X.

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