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.(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Uniform Convergence problem

**Physics Forums | Science Articles, Homework Help, Discussion**