(From Royden 3rd Ed )How to prove the delta-approximation converges to f in measure as delta goes to zero?

Question

In Royden 3rd Ed, its given how delta-approximation to a function f in L^P converges to f in L^P. Now I need to show how it converges "in measure". Im so stuck on it that I dont know how to go on. Plz help me! Im providing both the question and the other proof on how it converges in L^P.

this is the proof given for converge in L^P. I need help on proving how it converges in measure

Answer 1

Convergence in $L^{p}$ always implies convergence in measure: $\mu \{x:|T_{\Delta}f -f| >\epsilon\} \leq \frac 1 {\epsilon ^{p}} \int |T_{\Delta} f -f|^{p} d\mu \to 0$.