Let $\mu$ be a $\sigma$-finite measure and suppose that $(f_n)$ is a sequence (of real functions) in $L_\infty(\mu)$ that is bounded by some $g\in L_\infty(\mu)$ and is increasing. Clearly, $(f_n)$ converges to some function $f$ in the weak* topology of $L_\infty(\mu)$. Does it also converge in the weak topology, that is, is it true that for any $\varphi\in L_\infty(\mu)^*$ we have
$$\langle f_n,\varphi\rangle \to \langle f, \varphi\rangle?$$
That need not be. Let $\mu$ the counting measure on $\mathbb{N}$, and $f_n = \sum_{k = 0}^n e_k$ with $e_k$ the "standard" unit vectors in $\ell^{\infty}$. Let $\varphi$ any continuous extension of $\lambda \colon c \to \mathbb{C}$ given by $\lambda(f) = \lim\limits_{k\to\infty} f(k)$. Then we have $\varphi(f_n) = 0$ for all $n$, but $f_n \uparrow \mathbb{1}$, and $\varphi(\mathbb{1}) = 1$.