25.5 Measurable Functions into Metric Spaces

Proposition 25.5.1.label Let $(X, \cm)$, $(Z, \cn)$ be measurable spaces, $\seqf{Y_j}$ separable metrisable topological spaces, $F: \prod_{j = 1}^{n} Y_{j} \to Z$ be a $(\cb_{\prod_{j = 1}^n Y_j}, \cn)$-measurable function. For any $\seqf{f_j}$ where for each $1 \le j \le n$, $f_{j}: X \to Y_{j}$ is $(\cm, \cb_{Y_j})$-measurable, the composition

is $(\cm, \cn)$-measurable.

Proposition 25.5.2.label Let $(X, \cm)$ be a measurable space, $Y$ be a separable and metrisable topological space, $f, g: X \to Y$ be $(\cm, \cb_{Y})$-measurable functions, then the following functions are measurable:

1. (1)

   For any metric $d$ on $Y$, $x \mapsto d(f(x), f(y))$.

2. (2)

   If $Y$ is a TVS over $K \in \RC$ and $\lambda \in K$, $\lambda f + g$. In particular, $\bracs{f = g}\in \cm$.

3. (3)

   If $Y \in \RC$, $fg$.

Proposition 25.5.3.label Let $(X, \cm)$ be a measurable space, $Y$ be a metrisable topological space, and $f: X \to Y$ be a function, then the following are equivalent:

1. (1)

   $f$ is $(\cm, \cb_{Y})$-measurable.

2. (2)

   For each $\phi \in C(X; [0, 1])$, $\phi \circ f$ is $(\cm, \cb_{\real})$-measurable.

Proposition 25.5.4.label Let $(X, \cm)$ be a measurable space, $Y$ be a metrisable topological space, and $\seq{f_n}$ be $(\cm, \cb_{Y})$-measurable functions, then:

1. (1)

   If $Y$ is Polish, then $\bracsn{\limv{n}f_n \text{ exists}}\in \cm$.

2. (2)

   If $f = \limv{n}f_{n}$ exists, then it is $(\cm, \cb_{Y})$-measurable.