# Weak convergences of measurable functions and of measures

My question is "how weak convergences of measurable functions is defined?" There seems to be two different definitions which are both based on weak convergence of measures generated by the measurable functions, but differ in how the measurable functions generate their new measures.

1. In terms of the new measure $\nu(A):=\int_A f \,d\mu$ generated from a measurable function $f$ wrt a measure $\mu$ on its domain:

From "Weak Compactness" in Section 19 "The $L^p$ spaces" of "Probability and Measure" by Billingsley:

Suppose that $f$ and $f_n$ are elements of $L^p(\Omega, \mathcal{F},\mu)$. If $\int f \times g \, d\mu= \lim_{n\rightarrow \infty} \int f \times g \, d\mu$ for each $g$ in $L^q(\Omega, \mathcal{F},\mu)$ with $1/p + 1/q = 1$, then $f_n$ converges weakly to $f$.

At the end of Section 4.1 of "A Course in Probability Theory" by Kai Lai Chung, weak convergence of random variables in $L^1$ space is also defined similarly to Billingsley's.

2. In terms of the pushforward measure of a measurable function:

In Wikipedia,

In this case the term weak convergence is preferable (see weak convergence of measures), and we say that a sequence of random elements $\{X_n\}$ converges weakly to $X$ if $$\operatorname{E}^*h(X_n) \to \operatorname{E}\,h(X)$$ for all continuous bounded functions $h(·)$. Here $E^*$ denotes the outer expectation, that is the expectation of a “smallest measurable function g that dominates $h(X_n)$”.

Also from Wikipedia:

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $\textbf{X}$ be a metric space. If $X_n, X: Ω → \textbf{X}$ is a sequence of random variables then $X_n$ is said to converge weakly (or in distribution or in law) to $X$ as $n → ∞$ if the sequence of pushforward measures $(X_n)_∗(P)$ converges weakly to $X_∗(P)$ in the sense of weak convergence of measures on $\textbf{X}$.

I wonder if the two definitions of weak convergence of measurable functions are equivalent? Why are there two different definitions for the same concept?

Thanks and regards!

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 $M_1\subset(C_b)^*$.
• Thanks! I am slow to understand your reply. Would you mind explaining what is the continuous dual of $C_b$? I opened a new post here math.stackexchange.com/questions/309783/…. Thanks! – Tim Feb 22 '13 at 14:58