Given that the sequence of $\mathcal{S}$-measurable functions $\{f_k\}_{k\geq1}$ are non-negative, for each $f_k$, we can construct an increasing sequence of simple non-negative functions $\{f_{k_n}\}_{n\geq1}$ such that $\lim_{n\to\infty}f_{k_n}(x)=f_k(x)$ for all $x\in\,X$. Now, by the monotone convergent theorem, we have
$$\sum_{k=1}^{m}\int\,f_kd\mu=\lim_{n\to\infty}\sum_{k=1}^{m}\int\,f_{k_n}d\mu=\lim_{n\to\infty}\int\sum_{k=1}^{m}f_{k_n}d\mu=\int\sum_{k=1}^{m}f_{k}d\mu$$
As $m$ goes to infinity, we have
Actually, I need to deduce something about the direction of a vector field if it's curl is known completely.
Let's say we have a vector field $\mathbf B$ which depends only on the distance $r$ (perpendicular to z-axis) and points in positive $z$ direction. [I apologize for not being able to pu...
You have $T^2(V)\subseteq T(V)$, and the dimension of these two subspaces is equal so $T(V) = T^2(V)$. Hence, $T$ is 1-1 on $T(V)$.
