Thread: Modulus of continuity is continuous?

Hey,

I'm currently reading a book on convergence of probability measures, and there is a property that they assert without too many details that I can't manage to work out for myself.

To put you in context, we're in the space of functions that are continuous with respect to the standard euclidean metric , and we define the metric on to be .

For every and , define the modulus of continuity as




In the book, they say that for any fixed , the function is continuous. Their only argument is that
for any , we have . It's easy to see how this implies continuity, but I can't manage to show this inequality myself. Any help or hints would be greatly appreciated.

We have then take supremum. Switch the roles of and to get what you want.

P.S. Which book are you studying?

Thanks for the answer, I'll try this out.

I'm studying Convergence of Probability Measures 2nd edition by Patrick Billingsley, I'm trying to gain a better understanding of Donsker's Theorem, which is basically a version of the Central Limit Theorem for random variables taking values in C[0,1].