---- > [!definition] Definition. ([[weak convergence of measures|weak convergence of measures]]) > - Let $X$ be a [[topological space]]. > - Let $C_{b}(X)$ denote the collection of [[continuous]] [[bounded set|bounded]] "test" functions $X \to \mathbb{R}$ > - Let $\mathcal{M}(X)$ be a collection of [[finite measure|finite]] [[measure|measures]] on $X$.[^1] > - For each $f \in C_{b}(X)$, define $\begin{align} > T_{f}: \mathcal{M}(X)& \to \mathbb{R} \\ > \mu & \mapsto \int f \, d\mu . > \end{align}$ > > A [[sequence]] $\mu_{1},\mu_{2},\dots$ of [[measure|measures]] is said to **weakly (or initially) converge** to $\mu \in \mathcal{M}(X)$ if it [[converge|converges]] to $\mu$ in the [[initial topology]] on $\mathcal{M}(X)$ wrt $\{ T_{f} \}_{f \in C_{b}(X)}$. > [!equivalence] > Unwinding the characterization of [[initial topology]] as [[initial topology|initial topology is topology of pointwise convergence]], we see $\mu_{n} \to \mu \text{ weakly }\iff \int f \, d\mu_{n} \to \int f \, d\mu \text{ for all } f \in C_{b}(X).$ ^equivalence > [!definition] Definition. (Convergence of pushforwards/convergence in distribution) > Let $(A, \Sigma, \nu)$ be any [[measure|measure space]]. A [[sequence]] of functions $g_{1},g_{2},\dots:A \to X$ is said to **converge in distribution** to $g:A \to X$ if the [[pushforward measure|pushforwards]] $(g_{1})_{*}\nu, (g_{2})_{*}\mathbf{\nu}\dots$ weakly converge in measure to $g_{*}\nu$. The name is motivated the special case of probability: [[convergence in distribution of random variables]]. > > [[Convergence in measure]] implies convergence in distribution for finite measures.[^1] [^1]: Must show $(g_{n})_{*} \nu \to g_{*}\nu$, i.e., that the [[integral|integrals]] $\underbrace{ \int_{X} f\, d(g_{n})_{* }\nu }_{ = \int _{A} f \circ g_{n} \, d\nu } \to \underbrace{ \int _{X} f \, dg_{*}\nu }_{ = \int _{A} f \circ g \, d\nu } $ [[integral with respect to a pushforward measure|for]] all $f \in C_{b}(X)$. Since [[continuous|continuity]] preserves [[convergence in measure]], $f \circ g_{n} \to f \circ g$ in measure. Put $h_{n}=f \circ g_{n}$ and $h=f \circ g$. So just need to justify exchanging the limit and integral. Since the measure is finite, there exists a [[subsequence]] $h_{n_{k}} \to h$ a.e. Apply [[Dominated Convergence Theorem|DCT]] to get $\int _{A} h_{n_{k}} \, d\mu \to \int _{A} h \, d\mu$. Then justify upgrading to full sequence (TODO) [^1]: With the [[σ-algebra]] on the [[topological space]] $X$ being, as usual, the [[Borel set|Borel σ-algebra]] $\mathcal{B}(X)$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```