What is the $\sigma(L^1,L^\infty)$-topology?

Question

In chapter 4, paragraph 6 of Revuz' Markov chains book, he's considering a sequence $(s_n)_{n\in\mathbb N}$ of $L^1$-functions on a measure space. In Proposition 6.6 (see below), he's talking about limit points of $(s_n)_{n\in\mathbb N}$ in the "$\sigma(L^1,L^\infty)$"-topology. What kind of topology is this? I can't find the definition of it inside the book.

Answer 1

$\sigma(X,X^*)$ is often used to indicate the weak topology on the Banach space $X$ (and $\sigma(X^*,X)$ indicates the weak-* topology on $X^*$). $L^\infty$ should be identified with $(L^1)^*$ via the Riesz isomorphism $f\mapsto \int (\bullet)\cdot f\,d\mu$.

Answer 2

That one is the "Weak Topology", the weakest topology which make every element of $L^{1*}$ continuos. In this case we know $L^{1*}=L^{\infty}$.