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.

Re: Modulus of continuity is continuous?

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

P.S. Which book are you studying?

Re: Modulus of continuity is continuous?

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].