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Lebesgue Integration and Measure Theory

(R,B,λ) denote the real line with Lebesgue measure defined on the Borel subsets of R.
And 1&#8804;p<&#8734;

1. Show that the sequence fn = n &#967;[1/n, 2/n] (&#967; is a step function) has property that if &#948;>0, then it is uniformly convergent on the complement of the set [0,&#948;]. However, show that there does not exist a set of measure zero, on the complement of which (fn) is uniformly convergent.

2. Show that fn = n &#967;[1/n, 2/n] converges almost uniformly but not in Lp(R,B,&#955;).

3. Show that fn = &#967;[n, n+1] converges everywhere, but not almost uniformly.


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