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Lebesgue Measure

6.T. Show that if f∈L∞(X,X,µ), then |f(x)|≤∥f∥∞ for almost all x. Moreover,
if A <&#8741;f&#8741;&#8734;then there exists a set E&#8712;X with µ(E)>0 such that |f(x)|>A for all x&#8712;E.

6.U. Show that if f&#8712;Lp, 1&#8804;p&#8804;&#8734;, and g&#8712;L&#8734;, then the product fg&#8712;Lp and
&#8741;fg&#8741;p &#8804; &#8741;f&#8741;p&#8741;g&#8741;&#8734;

From The Elements of Integration and Lebesgue Measure written by R.G. Bartle

6.T. and 6.U. (attached)

See attached file for full problem description.

keywords: integration, integrates, integrals, integrating, double, triple, multiple, integrable, integrability, measurable, measurability

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A Lebesgue measure is investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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