Everyone I'm just doing an intro to measure theory right now, Durrett Chapter 1,(we are using Lebesgue measure) and am trying to get hold of the different types of convergence. Right now, I know of only 4 types: pointwise convergence, uniform convergence, convergence almost everywhere, convergence in measure.
I know that uniform convergence implies pointwise convergence and that pointwise convergence implies convergence almost everywhere. I also know of an example of a sequence of functions that converges in measure but does not converge almost everywhere.
Thus, my question does uniform convergence necessarily imply convergence in measure? Or is there a specific example where a sequence of functions does converge uniformly but does not converge in measure.
Thank you.
On a probability space if a sequence of random variable $(X_n)$ converge uniformly, then they converge in measure (or else called probability). Indeed if $\lVert X_n -X \rVert_{\infty}\to 0$, then $$ E|X_n-X|\le \lVert X_n -X \rVert_{\infty}\to 0 $$ so $X_n\to X$ in probability by Markov's inequality. Indeed, $$ P(|X_n-X|>\epsilon)\leq\frac{E|X_n-X|}{\epsilon}\to0 $$ for any $\epsilon >0$.