Let \((X,\mathfrak M,\mu )\) be a positive measure space, and let \(f\in L^1(X)\). Let \(X_n=\{x\in X: |f(x)|>n\}\). Show that \[ \lim _{n\to \infty } n\mu (X_n)=0. \]
Question
Math 6210 • Real Analysis
2004 • Question 1
Question
2004 • Question 1
Let \((X,\mathfrak M,\mu )\) be a positive measure space, and let \(f\in L^1(X)\). Let \(X_n=\{x\in X: |f(x)|>n\}\). Show that \[ \lim _{n\to \infty } n\mu (X_n)=0. \]