Weak convergences of measurable functions and of measures

In the chapter you quote, Billingsley defines weak convergence of functions in $L^p$, which is by definition the convergence against every function in $L^q$, the dual of $L^p$. That is, $(f_n)$ in $L^p$ converges to $f$ in $L^p$ if and only if $\int f_ng$ converges to $\int fg$ for every $g$ in $L^q$ and this is a weak convergence because $L^q=(L^p)^*$.

In the chapter you quote, Chung defines weak convergence of functions in $L^1$, which is by definition the convergence against every function in $L^\infty$, the dual of $L^1$. That is, $(f_n)$ in $L^1$ converges to $f$ in $L^1$ if and only if $\int f_ng$ converges to $\int fg$ for every $g$ in $L^\infty$ and this is a weak convergence because $L^\infty=(L^1)^*$.

In the page you quote, Wikipedia defines weak convergence of (probability) measures. Actually, this mode of convergence should be called weak* rather than weak (but usage is to call it weak) because it refers to the convergence against any bounded continuous functions, and the space $M_1$ of probability measures is included in the dual of the space $C_b$ of bounded continuous functions. That is, $(\mu_n)$ in $M_1$ converges to $\mu$ in $M_1$ if and only if $\int f\mathrm d\mu_n$ converges to $\int f\mathrm d\mu$ for every $f$ in $C_b$ and this is a weak* convergence because $M_1\subset(C_b)^*$.

To mean that the distributions of some random variables converge weakly in the sense above, one usually says that the random variables converge in distribution.