To define addition and multiplication operations, we need the following concepts. the function f : N →N and the following result. It holds at least one of the statements (by the method of induction with fixed b). Then exactly one of the following three statements is true: ii) x is equal to a positive natural number,. iii) x is the negation of a positive natural number.
In the trichotomy of natural numbers, exactly one of the following statements applies: a = b, a > b, a < b. Let x ∈ Q. Then exactly one of the following three statements is true:. ii) x is a positive rational number, (iii) x is a negative rational number. Clearly, it cannot be more than one of the three statements that holds at the same time. ii) If a and b are both positive, then x is positive by the definition of positivity of rational number, directly.
A Cauchy sequence {an}∞n=1 is called non-degenerate if there is a rational number c > 0 such that |an| ≥ c for all n∈ N. The last proposition shows that every nonzero real number can be expressed by a nondegenerate Cauchy sequence. Theorem. (Trichotomy of real numbers) Let a ∈ R. One of the following three statements is true:. The equivalence of (i) and (ii) is a direct consequence of defining the positivity of the real numbers.
The equivalence of (i) and (ii) is a direct consequence of the definition of positivity of real numbers. iii) ⇒ (iv) Assume that statement (iii) holds.
Theory of sets
Let F be a class of subsets of a set X. The following statements are equivalent. ii) F is closed under operations of union, difference and complement. iii). A in disjoint equivalence classes, that is, Ais a disjoint union of the equivalence classes under ∼. Partial order). Let (A,≺) be a partially ordered set, and B a subset of A. Suppose that C is a set of nonempty sets.
Lie's concept of a continuous group of transformations without the assumption of the differentiation of the functions defining the group. Impossibility to solve the general equation of the 7th degree using functions of only two arguments. Since A is an open interval, we can select a rational number rA ∈ A based on the density of the rational numbers.
If a subset of R is both open and closed, then it is either ∅ or R. The closure of a set A⊂ R is closed. The sequence {an} is unique unless x 6= 1 and is of the form pqm for some q, m ∈ N, in which case there are exactly two such sequences, one with only finitely many nonzero terms and the other with only finitely many terms differ from p−1. Since the Cantor set is defined as the set of points that are not excluded, the proportion (i.e., Lebesgue measure) of the unit interval remaining can be found by removing total length.
In fact, it may seem surprising that anything remains—after all, the sum of the lengths of the removed intervals equals the length of the original interval. It may seem that there are "only" endpoints left, but this is not the case either. For example, the number 1/4 is in the bottom third, so it is not removed in the first step, and it is in the top third of the bottom third, and it is in the bottom third of "this" and in the "top" third of "this", and so on ad infinitum; alternating between the upper and lower thirds.
Because it is never in any of the middle thirds, it is never removed, and yet neither is it one of the endpoints of any middle third. In the sense of cardinality, 'most' members of the Cantor set are not endpoints of removed intervals. Thus, the middle third (which must be removed) contains the ternary digits of the form.
The set of endpoints of the deleted intervals is countable, so there must be uncountable numbers in the Cantor set that are not interval endpoints. These, among other properties of the Cantor set, make it useful for illustrating many subtle concepts (for example, the so-called 'measure').
Theory of measures
The outer measure of a measurable Lebesgue group is called E, and we write it by m(E). It is enough to show that m. Clearly, from the additivity and finite monotonicity of m, it follows that. Thus, it suffices to show the equivalence of (i) and (ii). i) ⇒(ii) Assume that E is Lebesgue measurable.
We call a Lebesgue measurable function anE if f−1(O) ∈ M for every open set O ⊂R. Then E ∈ M if and only if χE is Lebesgue measurable in R. Indeed, this is a direct consequence of the following fact:. Since Oc is closed for any open set O, and Fc is open for any closed set F, we obtain the conclusion from the definition of measurable Lebesgue functions immediately. The following statements are equivalent. ii) f|E1 and f|E2 are measurable Lebesgue functions on E1 and E2, respectively. i) ⇒(ii) Suppose f is Lebesgue measurable in E .
Let f and g be two Lebesgue measurable functions on E ⊂ R. i) αf is Lebesgue measurable for every α ∈ R. vi) max{f, g} and min{f, g} are both Lebesgue measurable functions. If limn→∞fn is finite, then limn→∞fn is Lebesgue measurable. v) If {fn} converges pointwise to a finite real function f, then f is Lebesgue measurable. A function f : E ⊂ R →R is said to be a simple function if f is a linear combination of characteristic functions of Lebesgue measurable sets, i.e. there exist constants α1, α2,· · · , αN and Lebesgue measurable sets E1, E2, · · · , EN so that.
A Lebesgue measurable function f is a simple function if and only if the range of f is a finite set. It is clear that fn is a simple function since all sets En,k are Lebesgue measurable. In the last two approximation theorems we can choose the desired sequence {fn} of such simple functionsfn, where each fn is a linear combination of characteristic functions of Lebesgue measurable sets with finite measure.
All continuous functions, step functions, simple functions and monotone functions are Lebesgue measurable functions. With the last theorem we immediately obtain the following approximation theorem for Lebesgue's measurable functions. For measurable Lebesgue functions we have the following result thanks to Nikolai Lusin (or Luzin-Russian).
If f is Lebesgue measurable on E, then for every > 0 there exists a closed subset F ⊂ E with m(E\F) < such that f|F is continuous on F. The following Littlewood three principles provide a useful intuitive guide in the initial study of the theory of Lebesgue measurable sets and measurable functions: i) Every measurable set is almost a finite union of intervals. ii) Every measurable function is nearly continuous. iii) Every convergent sequence of measurable functions is almost uniformly convergent.
Lebesgue integrals
By the Heine theorem, f is Lebesgue integrable on E if and only if the limit limn→∞S(Dn) exists for every sequence {Dn} of partitions of [m, M) with δ(Dn) → 0 as n → ∞ and is independent of D and ηk. Thus the integral of f is only dependent if the value a and b, then independent of m and M. ii) The Dirichlet function D is not Riemann integrable on [0,1], while Lebesgue integrable on [0,1] with . This is a direct consequence of the theorem above once we have observed it. by the definition of integrals.
From this point of view, we can observe that the Dirichlet function is integrable on [0,1], immediately. Then f is Lebesgue integrable and by the finite additivity of the Lebesgue integral we have. We call f is Lebesgue integrable on E if the limit s := lim. and call s the Lebesgue integral of f on E, which is denoted by . i) It is clear that the integral of f on E defined above is unique, that is, s is independent of the choice of {En} and {Mn}.
A Lebesgue measurable function f onE is said to be Lebesgue integrable on E if f+ and f− are both Lebesgue integrable on E. Then f is Lebesgue integrable on E if and only if f is Lebesgue integrable on both E1 and E2. Linear property) Let f and g be two Lebesgue-integrable functions on E. Then for all a, b ∈ R, af +bg is Lebesgue-integrable on E and.
From (vi) above, we observe that f is integrable on E if and only if |f| is integrable on E, but not for generalized Riemann integrals. Thus, the new integral is an extension of the true Riemann integral (or the improper Riemann integral of nonnegative functions), not the general improper Riemann integral. Complete continuity) Let f be integrable on E. Let f and g be integrable functions on E ⊂ R. Recall that the linear integral property reads that.
Let E ⊂ R be a measurable set and {fn} be a set of measurable functions on E, and let F be a non-negative measurable function on E. Lebesgue dominated convergence) Let {fn} be a set of measurable functions on E domi - characterized by an integrable functionF. Then, by step I, {gn} converges almost everywhere to an integrable g on E, or equivalently, {fn} converges almost everywhere to an integrable f on E, and.
