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Assume that 0 < m(E) < ∞ and (fn)n converges pointiwise to f, a.e. For > 0, consider the sequence of sets: A₁ = {\fn-f|> }, n=1,2,... (1) Show that, for every e > 0, XA converges pointwise to 0, a.e., as n → +∞ (2) Show that (fn)n converges to f in measure. (Hint: use Lebesgues Dominated Convergence Theorem). (3) By means of the counter-example E = R and fn = X(n,n+1), check that the result of (2) no longer holds true if m(E) = ∞.

Assume that 0 < m(E) < ∞ and (fn)n converges pointiwise to f, a.e. For > 0, consider the sequence of sets: A₁ = {\fn-f|> }, n=1,2,... (1) Show that, for every e > 0, XA converges pointwise to 0, a.e., as n → +∞ (2) Show that (fn)n converges to f in measure. (Hint: use Lebesgues Dominated Convergence Theorem). (3) By means of the counter-example E = R and fn = X(n,n+1), check that the result of (2) no longer holds true if m(E) = ∞.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 5 Assume that 0 < m(E) <∞ and (fn)n converges pointiwise to f, a.e. For > 0, consider the sequence of sets: A₁ = {|fn-f|> ε}, n = 1, 2,... (1) Show that, for every > 0, XA converges pointwise to 0, a.e., as n → +∞ (2) Show that (fn)n converges to f in measure. (Hint: use Lebesgues Dominated Convergence Theorem). (3) By means of the counter-example E = R and fn = X(n,n+1), check that the result of (2) no longer holds true if m(E) = ∞.
Transcribed Image Text:Question 5 Assume that 0 < m(E) <∞ and (fn)n converges pointiwise to f, a.e. For > 0, consider the sequence of sets: A₁ = {|fn-f|> ε}, n = 1, 2,... (1) Show that, for every > 0, XA converges pointwise to 0, a.e., as n → +∞ (2) Show that (fn)n converges to f in measure. (Hint: use Lebesgues Dominated Convergence Theorem). (3) By means of the counter-example E = R and fn = X(n,n+1), check that the result of (2) no longer holds true if m(E) = ∞.
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