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futurebird
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I'm trying to better understand convergence so I made upa problem for myself based on an example from class. I want to know if I'm answering my own questions correctly.
Define a sequence of functions fn(x) = 1 if x is in {r1, r2, ... , rn} and 0 otherwise. Where r1, r2, ... , rn are the first n rational numbers in some enumeration of all rational numbers. fn converges pointwise to the dirichlet function. But, can we say anything else about how fn --> f?
Uniform Convergence
Given e > 0 is there an N such that when n > N |fn - f | < e for all x? No. Just let e=.5 we cal alays find an x value where |fn - f | = 1. That is, a rational number that has not yet been listed by the time we reach n.
Convergence in Measure
Yes. The measure of the set where f and fn are not the same is *always* 0.
Almost Uniform Convergence
Yes. If we let A, the set of measure less the any e where uniform convergence fails be Q, the rationals I think we have almost uniform convergence. since m{Q}=0 < e for all e > 0.
Convergence in LP
Yes, the Lp norm of the fn and f is always 0 anyway.
Define a sequence of functions fn(x) = 1 if x is in {r1, r2, ... , rn} and 0 otherwise. Where r1, r2, ... , rn are the first n rational numbers in some enumeration of all rational numbers. fn converges pointwise to the dirichlet function. But, can we say anything else about how fn --> f?
Uniform Convergence
Given e > 0 is there an N such that when n > N |fn - f | < e for all x? No. Just let e=.5 we cal alays find an x value where |fn - f | = 1. That is, a rational number that has not yet been listed by the time we reach n.
Convergence in Measure
Yes. The measure of the set where f and fn are not the same is *always* 0.
Almost Uniform Convergence
Yes. If we let A, the set of measure less the any e where uniform convergence fails be Q, the rationals I think we have almost uniform convergence. since m{Q}=0 < e for all e > 0.
Convergence in LP
Yes, the Lp norm of the fn and f is always 0 anyway.