I was looking at some exercise to prepare for my exam and I found a couple of questions that I couldn't solve. I would love some hints or see how it should be done.
1) Let $(X, \mathscr{A}, \mu)$ be a probability space and $u\in \mathcal{M}^+(\mathscr{A})$. Show that $$\left| \int_X ud\mu\right|\leq||u||$$ Where $\mathcal{M}^+(\mathscr{A})$ are the $\mathscr{A}$-measurable positive functions and $||u||$ is the uniform norm of $u$.
2) For $(\mathbb{R}, \mathcal{B}(\mathbb{R}), \lambda )$ find a sequence in $\mathcal{M}^+(\mathscr{\mathcal{B}(\mathbb{R})})$ such that $v_n \rightarrow 0$ for $n \rightarrow \infty$ uniformly but $ \lim_{n \rightarrow \infty} \int_\mathbb{R} v_n d\lambda \neq 0 $
3) Let $(X, \mathscr{A}, \mu)$ be a probability space, $u\in \mathcal{M}^+(\mathscr{A})$ and $u_n$ be a sequence in $\mathcal{M}^+(\mathscr{A})$. Assume $u_n \rightarrow u$ for $n \rightarrow \infty$ uniformly. Show that $$ \lim_{n \rightarrow \infty} \int_X u_n d\mu=\int_Xud\mu$$
I'm not sure where to start on the first two but in the last one I think that I could use a theorem saying that I can rewrite $u_n$ to an increasing sequence of simple functions for which I can use a rule stating that I can move the lim in the integral and by that take the limit of the simple functions yielding $u$
I hope you can help, thanks in advance!
For a non-negative simple function $u$ 1) follows immediately from definition and the general case follows by writing $u$ as the limit of an increasing sequence of simple functions $\{u_n\}$. 3) is immediate from 1). For 2) take $u_n=\frac 1 n I_{(0,n)}$ Details for 1): if $u=\sum_{k=1}^{N} c_k I_{E_{k}}$ with $c_k \geq0$ and $E_k$'s disjoint then $\int u \, d\mu =\sum_{k=1}^{N} c_k \mu({E_{k}})\leq \max \{c_k: 1\leq k \leq N\}=\|u\|$. [ Note that absolute value sign in 1) is unnecessary because the integral is non-negative]. Let $u$ be any non-negative simple function. There is a sequence $\{u_n\}$ of non-negative simple functions incresing to $u$. Note that $0\leq u_n \leq u$ so $\|u_n\|\leq \|u\|$ for all $n$. Hence $\int u_n\, d\mu \leq \|u_n\|\leq \|u\|$ for all $n$. Take limit as $n \to \infty$ and use Monotone Convergence Theorem to complete the proof of 1.