A Short Course on the Lebesgue Integral and Measure Theory by Steve Cheng

By Steve Cheng

This text develops the fundamentals of the Lebesgue imperative and degree conception. when it comes to content material, it provides not anything new to any of the present textbooks at the topic. yet our technique the following may be to prevent unduly abstractness and absolute generality, in its place concentrating on generating proofs of necessary effects as quick as attainable. a lot of the fabric right here comes from lecture notes from a brief actual research direction I had taken, and the remaining are famous effects whose proofs I had labored out myself with tricks from a variety of resources. I typed this up regularly for my very own gain, yet i'm hoping will probably be fascinating for a person all for the Lebesgue fundamental (or greater arithmetic in general). i'll be offering proofs of each theorem. while you're bored interpreting them, you're invited to do your individual proofs. The bibliography outlines the history you must comprehend this text.

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Example text

It is even true that strong sigma-compactness implies local compactness in a metric space. 10. Let X be a strongly sigma-compact metric space, and µ be any locally finite measure on B(X). Then the space of continuous functions with compact support is dense in Lp (X, B(X), µ), 1 ≤ p < ∞. 15 Other examples of measures Since so far we have chiefly worked only in Rn with Lebesgue measure, it should be of interest to give a few more useful examples of measures. k-dimensional volume of a k-dimensional manifold A manifold is a generalization of curves and surfaces to higher dimensions, and sometimes even to spaces other than Rn .

Countable additivity requires the typical approximation arguments. Suppose that we have Jn ∈ A with infinite disjoint union I ∈ A. By monotonicity we ∞ automatically have n=1 µ(Jn ) ≤ µ(I), so we only have to prove the other inequality. We can assume that Jn are simple intervals (an , bn ], instead of finite disjoint unions of intervals. Also assume, for now, that I is the finite interval (a, b]. Since F is right-continuous, for every ε > 0, there exists δ > 0 such that 0 ≤ F (b) − F (a) < F (b) − F (a + δ) + ε .

Given f : X → Rn measurable, let {ek } denote the standard basis vectors in Rn , and {f k } the components of f with respect to this basis. Of course we define n f k ek , f= k=1 provided the integrals on the right exist. Generally, to say that X f exists, we do not allow any one of the components to be infinite, for this is usually not useful when n ≥ 2. It is not hard to see that f is measurable if and only if each f k is. It follows immediately that this integral is linear, and hence, the definition is independent of the basis, as it ought to be.

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