I need to prove the following:
If $\alpha$ is a probability measure and ${X_n \to X} $ a.e then ${X_n \to X}$ in measure. Show that the opposite may not be true.
I need to prove the following:
If $\alpha$ is a probability measure and ${X_n \to X} $ a.e then ${X_n \to X}$ in measure. Show that the opposite may not be true.
For simple, let $Y_n:=|X_n-X|$. We have $P\left(\bigcap_{l=1}^{+\infty}\bigcup_{n\geq 1}\bigcap_{j\geq n}\{Y_j\leq l^{—1}\}\right)=1$ by hypothesis, hence $P\left(\bigcup_{l=1}^{+\infty}\bigcap_{n\geq 1}\bigcup_{j\geq n}\{Y_j>l^{—1}\}\right)= 0.$ This gives that for each $l\geq 1$, $P(\limsup_{j\to +\infty}\{Y_j>l^{-1}\})=0$. This implies $\limsup_{j\to +\infty}P(\{Y_j>l^{-1}\})=0$ for each $l$, what we wanted.
For a counter-example showing the converse doesn't hold, see this thread.