Informally, a measure has the property of being monotone in the sense that if is a subset of the measure of is less than or equal to the measure of Furthermore, the measure of the empty set is required to be 0.
If are measurable sets that are increasing (meaning that ) then the union of the sets is measurable and
Continuity from aboveEdit
If are measurable sets that are decreasing (meaning that ) then the intersection of the sets is measurable; furthermore, if at least one of the has finite measure then
This property is false without the assumption that at least one of the has finite measure. For instance, for each let which all have infinite Lebesgue measure, but the intersection is empty.
A measurable set is called a null set if A subset of a null set is called a negligible set. A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called complete if every negligible set is measurable.
A measure can be extended to a complete one by considering the σ-algebra of subsets which differ by a negligible set from a measurable set that is, such that the symmetric difference of and is contained in a null set. One defines to equal
Both and are monotonically non-increasing functions of so both of them have at most countably many discontinuities and thus they are continuous almost everywhere, relative to the Lebesgue measure.
If then so that as desired.
so that as required.
If for all then we are done, so assume otherwise. Then there is a unique such that is infinite to the left of (which can only happen when ) and finite to the right.
Arguing as above, when Similarly, if and then
For let be a monotonically non-decreasing sequence converging to The monotonically non-increasing sequence of members of has at least one finitely -measurable component, and
Continuity from above guarantees that
The right-hand side then equals if is a point of continuity of Since is continuous almost everywhere, this completes the proof.
Measures are required to be countably additive. However, the condition can be strengthened as follows.
For any set and any set of nonnegative define:
That is, we define the sum of the to be the supremum of all the sums of finitely many of them.
A measure on is -additive if for any and any family of disjoint sets the following hold:
Note that the second condition is equivalent to the statement that the ideal of null sets is -complete.
A measure space is called finite if is a finite real number (rather than ). Nonzero finite measures are analogous to probability measures in the sense that any finite measure is proportional to the probability measure A measure is called σ-finite if can be decomposed into a countable union of measurable sets of finite measure. Analogously, a set in a measure space is said to have a σ-finite measure if it is a countable union of sets with finite measure.
For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite. Consider the closed intervals for all integers there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real numbers with the counting measure, which assigns to each finite set of reals the number of points in the set. This measure space is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to the Lindelöf property of topological spaces.[original research?] They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'.
Strictly localizable measuresEdit
Let be a set, let be a sigma-algebra on and let be a measure on We say is semifinite to mean that for all 
Semifinite measures generalize sigma-finite measures, in such a way that some big theorems of measure theory that hold for sigma-finite but not arbitrary measures can be extended with little modification to hold for semifinite measures. (To-do: add examples of such theorems; cf. the talk page.)
Every sigma-finite measure is semifinite.
Assume let and assume for all
We have that is sigma-finite if and only if for all and is countable. We have that is semifinite if and only if for all 
Taking above (so that is counting measure on ), we see that counting measure on is
sigma-finite if and only if is countable; and
semifinite (without regard to whether is countable). (Thus, counting measure, on the power set of an arbitrary uncountable set gives an example of a semifinite measure that is not sigma-finite.)
Let be a complete, separable metric on let be the Borel sigma-algebra induced by and let Then the Hausdorff measure is semifinite.
Let be a complete, separable metric on let be the Borel sigma-algebra induced by and let Then the packing measure is semifinite.
The zero measure is sigma-finite and thus semifinite. In addition, the zero measure is clearly less than or equal to It can be shown there is a greatest measure with these two properties:
Theorem (semifinite part) — For any measure on there exists, among semifinite measures on that are less than or equal to a greatest element
We say the semifinite part of to mean the semifinite measure defined in the above theorem. We give some nice, explicit formulas, which some authors may take as definition, for the semifinite part:
Let be uncountable, let be a -algebra on let be the countable elements of and let It can be shown that is a measure.
Measures that are not semifinite are very wild when restricted to certain sets.[Note 1] Every measure is, in a sense, semifinite once its part (the wild part) is taken away.
— A. Mukherjea and K. Pothoven, Real and Functional Analysis, Part A: Real Analysis (1985)
Theorem (Luther decomposition) — For any measure on there exists a measure on such that for some semifinite measure on In fact, among such measures there exists a least measure Also, we have
We say the part of to mean the measure defined in the above theorem. Here is an explicit formula for :
Results regarding semifinite measuresEdit
Let be or and let Then is semifinite if and only if is injective. (This result has import in the study of the dual space of .)
Let be or and let be the topology of convergence in measure on Then is semifinite if and only if is Hausdorff.
(Johnson) Let be a set, let be a sigma-algebra on let be a measure on let be a set, let be a sigma-algebra on and let be a measure on If are both not a measure, then both and are semifinite if and only if for all and (Here, is the measure defined in Theorem 39.1 in Berberian '65.)
Localizable measures are a special case of semifinite measures and a generalization of sigma-finite measures.
Let be a set, let be a sigma-algebra on and let be a measure on
Let be or and let Then is localizable if and only if is bijective (if and only if "is" ).
A measure is said to be s-finite if it is a countable sum of bounded measures. S-finite measures are more general than sigma-finite ones and have applications in the theory of stochastic processes.
For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instance, a countably additive set function with values in the (signed) real numbers is called a signed measure, while such a function with values in the complex numbers is called a complex measure. Observe, however, that complex measure is necessarily of finite variation, hence complex measures include finite signed measures but not, for example, the Lebesgue measure.
Another generalization is the finitely additive measure, also known as a content. This is the same as a measure except that instead of requiring countable additivity we require only finite additivity. Historically, this definition was used first. It turns out that in general, finitely additive measures are connected with notions such as Banach limits, the dual of and the Stone–Čech compactification. All these are linked in one way or another to the axiom of choice. Contents remain useful in certain technical problems in geometric measure theory; this is the theory of Banach measures.
A charge is a generalization in both directions: it is a finitely additive, signed measure. (Cf. ba space for information about bounded charges, where we say a charge is bounded to mean its range its a bounded subset of R.)
^One way to rephrase our definition is that is semifinite if and only if Negating this rephrasing, we find that is not semifinite if and only if For every such set the subspace measure induced by the subspace sigma-algebra induced by i.e. the restriction of to said subspace sigma-algebra, is a measure that is not the zero measure.
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Royden, H.L.; Fitzpatrick, P.M. (2010). Real Analysis (Fourth ed.). Prentice Hall. p. 342, Exercise 17.8. First printing. Note that there is a later (2017) second printing. Though usually there is little difference between the first and subsequent printings, in this case the second printing not only deletes from page 53 the Exercises 36, 40, 41, and 42 of Chapter 2 but also offers a (slightly, but still substantially) different presentation of part (ii) of Exercise 17.8. (The second printing's presentation of part (ii) of Exercise 17.8 (on the Luther decomposition) agrees with usual presentations, whereas the first printing's presentation provides a fresh perspective.)
Shilov, G. E., and Gurevich, B. L., 1978. Integral, Measure, and Derivative: A Unified Approach, Richard A. Silverman, trans. Dover Publications. ISBN 0-486-63519-8. Emphasizes the Daniell integral.
Teschl, Gerald, Topics in Real and Functional Analysis, (lecture notes)
Tao, Terence (2011). An Introduction to Measure Theory. Providence, R.I.: American Mathematical Society. ISBN 9780821869192.