(R,B,λ) denote the real line with Lebesgue measure defined on the Borel subsets of R.
1. Show that the sequence fn = n χ[1/n, 2/n] (χ is a step function) has property that if δ>0, then it is uniformly convergent on the complement of the set [0,δ]. 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 χ[1/n, 2/n] converges almost uniformly but not in Lp(R,B,λ).
3. Show that fn = χ[n, n+1] converges everywhere, but not almost uniformly.
Lebesgue Integration and Measure Theory are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.