By Rene Erlin Castillo, Humberto Rafeiro

Introduces reader to fresh themes in areas of measurable functions

Includes element of difficulties on the finish of every bankruptcy

Content allows use with mixed-level classes

Includes non-standard functionality areas, viz. variable exponent Lebesgue areas and grand Lebesgue spaces

This publication is dedicated solely to Lebesgue areas and their direct derived areas. distinctive in its sole commitment, this e-book explores Lebesgue areas, distribution services and nonincreasing rearrangement. furthermore, it additionally offers with susceptible, Lorentz and the more moderen variable exponent and grand Lebesgue areas with massive element to the proofs. The publication additionally touches on simple harmonic research within the aforementioned areas. An appendix is given on the finish of the e-book giving it a self-contained personality. This paintings is perfect for lecturers, graduate scholars and researchers.

Topics

Abstract Harmonic Analysis

Functional research

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**Additional resources for An Introductory Course in Lebesgue Spaces**

**Sample text**

We are now in condition to introduce a norm in the Lebesgue space. 17. 5) f p = f L p := ⎝ | f | p dμ ⎠ , X whenever 1 ≤ p < +∞. 5) does not define a norm when p < 1, we can take f = χ[0,1/2] , 1 g = χ[1/2,1] and we see that we have a reverse triangle inequality in L 2 ([0, 1], L , m). 79. We now want to see if the product of two functions in some L p is still in L p . The following example shows us that this is not always true. 18. Consider the function f (x) = |x|−1/2 if |x| < 1, 0 if |x| ≥ 1.

1007/978-3-319-30034-4 3 43 44 3 Lebesgue Spaces f Note that if A = 0, / then 0 is a lower bound on A, and thus inf(A) ∈ R. Let α = ∞ < ∞, we state that α ∈ A. Notice that Eα = {x ∈ X : | f (x)| > α } = ∞ {x ∈ X : | f (x)| > α + 1/n} n=1 moreover, for each n the set {x ∈ X : | f (x)| > α + 1/n} ∈ A. e. e. it follows f ∞ = f∗ ∞ = sup | f ∗ (x)| = sup | f (x)|. 2. We define L∞ (X, A , μ ), called the set of essentially bounded functions, by L∞ (X, A , μ ) = f : X → R is an A -measurable function and f ∞ <∞ .

The dual space of 1 1 p = x which is is ∞ ≤ x p p→∞ ∞ , . ∞. ∞. Proof. For all x ∈ 1 , we can write x = ∑∞k=1 αk ek , where ek = (δk j )∞j=1 forms a Schauder basis in 1 , since n x − ∑ αk ek = (0, . . , 0, αn+1 , . ) k=1 n and n x − ∑ αk ek ∞ = k=1 1 ∑ αk ek k=n+1 →0 1 since the series ∑∞k=1 αk ek is convergent. Let us define T ( f ) = { f (ek )}k∈N , for all f ∈ ( 1 )∗ . Since f (x) = ∑k∈N αk f (ek ), then | f (ek ) ≤ f , since ek 1 = 1. In consequence, supk∈N | f (ek )| ≤ f , therefore { f (ek )}k∈N ∈ ∞ .