8 Show that the operator is invertible 7 Do we have $\lim_{n\to \infty }\int X_n dP=\int\lim_{n\to \infty }X_ndP$ 6 Show that $\lim\inf A_n\subset \lim\sup_n A_n$ 5 A $\lambda$-system $\mathcal{L}$ that is a $\pi$-system is automatically a $\sigma$-field. 3 Find the average value of $2x^2 + 5x + 2$ on the interval where $x \in [1,3]$.

### Reputation (2,526)

 +10 Show that $\lim\inf A_n\subset \lim\sup_n A_n$ +10 Bernstein tail bound to expectation +10 Showing that a metric space is discrete if and only if any function from it to another metric space is continuous +10 Show that the operator is invertible

### Questions (22)

 9 Example of product space isomorphic to sum of subspaces 6 variance of maximum 5 Covariance of maxima 4 Inner product equivalence 3 Limit of sum of independent simple functions

### Tags (83)

 44 probability × 38 10 integration × 7 25 real-analysis × 31 10 operator-theory × 2 24 measure-theory × 17 10 norm × 2 18 probability-theory × 21 8 functional-analysis × 11 14 elementary-set-theory × 5 8 statistics × 10