Theory and Methods of Statistics

Appendix A

Results From Analysis and Probability

A.1 Some Important Results in Integration Theory

Theorem A.1.1

Lebesgue Dominated Convergence

Let { f_n}be a sequence of integrable functions on $\mathcal{X}$. If

(i) ${lim}_{n\to \infty }{f}_n(x)=f(x)$ a.e. in $\mathcal{X}$, that is, for all $x\notin \mathcal{S}$ where ${\int }_{S}dx=0$, and

(ii) there is an integrable function g on $\mathcal{X}$ such that | f_n(x)|≤ g(x) for all ...