By John Franks

This e-book presents a student's first stumble upon with the thoughts of degree conception and practical research. Its constitution and content material mirror the assumption that tricky techniques might be brought of their least difficult and such a lot concrete varieties. regardless of using the notice ``terse'' within the name, this article may additionally were referred to as A (Gentle) creation to Lebesgue Integration. it's terse within the experience that it treats just a subset of these recommendations ordinarily present in a considerable graduate-level research path. The e-book emphasizes the incentive of those ideas and makes an attempt to regard them easily and concretely. particularly, little point out is made from basic measures except Lebesgue till the ultimate bankruptcy and a spotlight is restricted to $R$ instead of $R^n$. After developing the first rules and effects, the textual content strikes directly to a few functions. bankruptcy 6 discusses classical actual and intricate Fourier sequence for $L^2$ capabilities at the period and exhibits that the Fourier sequence of an $L^2$ functionality converges in $L^2$ to that functionality. bankruptcy 7 introduces a few options from measurable dynamics. The Birkhoff ergodic theorem is acknowledged with no evidence and effects on Fourier sequence from bankruptcy 6 are used to turn out that an irrational rotation of the circle is ergodic and that the squaring map at the complicated numbers of modulus 1 is ergodic. This publication is appropriate for a complicated undergraduate direction or for the beginning of a graduate path. The textual content presupposes that the scholar has had a regular undergraduate direction in actual research

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**Example text**

Proof. It is immediate from the deﬁnition that for c ∈ R the function cf is measurable when f is. Suppose f and g are measurable. , that for any a ∈ R the set Ua = {x | f (x) + g(x) > a} is measurable. Let {rn }∞ n=1 be an enumeration of the rationals. , if f (x0 ) + g(x0 ) > a, then f (x0 ) > a − g(x0 ). Since the rationals are dense there is an rm such that f (x0 ) > rm > a − g(x0 ). Hence, if we deﬁne Vm = {x | f (x) > rm } ∩ {x | g(x) > a − rm }, then x0 ∈ Vm . So every point of Ua is in some Vm .

2. (Borel sets). If C is a collection of subsets of R and A is the the smallest σ-algebra of subsets of R which contains all the sets of C, then A is called the σ-algebra generated by C. Let B be the σ-algebra of subsets of R generated by the collection of all open intervals. B is called the Borel σ-algebra and elements of B are called Borel sets. In other words, B is the collection of subsets of R which can be formed from open intervals by any ﬁnite sequence of countable unions, countable intersections, or complements.

If f is a regulated function on the interval I = [a, b], then, for any ε > 0, it can be uniformly approximated within ε by a step function. In particular, if εn = 1/2n , there is a step function gn (x) such that |f (x) − gn (x)| < εn for all x ∈ I. The regulated integral b b f (x) dx was deﬁned to be lim a gn (x) dx. a We deﬁne two other approximating sequences of step functions for f. Let un (x) = gn (x) + 1/2n and vn (x) = gn (x) − 1/2n . Then un (x) ≥ f (x) for all x ∈ I because un (x)−f (x) = 1/2n +gn (x)−f (x) ≥ 0 since |gn (x) − f (x)| < 1/2n .