let fRR be measurable and intergrable functionFor k0 set Akx

let f:RR be measurable and intergrable function.For k>0 set A_k={xR| |f(x)|>k}. Prove that lim(n)k m(A_k )=0. (m means measure) .

Solution

The limit willbe zero because

As function is measurable and integrable at for all R and set A_K is also the same and in limit we arre again multiplying K so they will get cancelled at n tends to infinity hence the limit will be zero.