Rudin Semicontinuity Exercise

[Tedjn]

Homework Statement

This is #22 of Chapter 2 in Rudin's Real and Complex Analysis 3rd Ed.

Suppose that X is a metric space, with metric d, and that f : X -> [0,infinity] is lower semicontinuous, f(p) < infinity for at least one p in X. For n = 1,2,3,..., x in X, define

$$g_n(x) = \inf\,\{\,f(p) + nd(x,p)\,:\,p \in X\,\}$$
​

and prove that

(i) |g_n(x) - g_n(y)| <= nd(x,y),
(ii) 0 <= g₁ <= g₂ <= ... <= f,
(iii) g_n(x) -> f(x) as n -> infinity, for all x in X.

Homework Equations

f is lower semicontinuous if for every real a, f^-1((a,infinity]) is open in X.

The Attempt at a Solution

This was a from a homework problem due in class earlier today, that just killed me to think about. (i) and (ii) were straightforward, and then I just could not figure out how to show that sup g_n = f. Because g_n is defined as an inf, what is stopping each subsequent g_n from increasing very very little? Perhaps the lower semicontinuity of f, which wasn't necessary for (i) or (ii)? Any help would be appreciated.

 

[mighty2000]

Hi there,

Thank you for sharing your thoughts on this problem. It seems like you have made some progress on parts (i) and (ii), which is great! To address your concern about part (iii), you are correct in thinking that the lower semicontinuity of f is key. In order to show that gn(x) approaches f(x) as n approaches infinity, we need to show that for any given x in X and any given epsilon > 0, there exists an n0 such that for all n >= n0, we have |gn(x) - f(x)| < epsilon.

To do this, we can use the lower semicontinuity of f to construct a sequence of points in X that approach x and have f-values that get closer and closer to f(x). Specifically, for each n, we can choose a point pn in X such that f(pn) <= gn(x) + epsilon/2n. Then, we can show that the sequence pn approaches x and that f(pn) approaches f(x) as n approaches infinity. This will give us the desired result.

I hope this helps! Let me know if you have any further questions or if you would like me to elaborate on any part of the solution. Keep up the good work!