Given two sets $\mathcal{X}$ and $\mathcal{Y}$, for example $\mathcal{X} \subseteq \mathbb{R}$ and $\mathcal{Y} \subseteq \mathbb{R}$, can we define a space which contains all possible measurable functions from $\mathcal{X}$ onto $\mathcal{Y}$ ?

For instance, the mathematical definition of such space is given as follows:

$\mathfrak{M}(\mathcal{X}; \mathcal{Y}):= \left\{ f(\cdot)\in \mathcal{Y}^{\mathcal{X}} \left| \begin{matrix} (\mathcal{X}, \mathscr{G}) \;\;\text{and} \;\; (\mathcal{Y}, \mathscr{H}) \; \text{are Sigma Algebra} \\ \mathscr{G} \subseteq 2^{\mathcal{X}} \; \text{and}\; \mathscr{H} \subseteq 2^{\mathcal{Y}} \;\;\text{and} \;\; f(\cdot) \; \text{is measurable} \end{matrix} \right. \right\} $

  • $\begingroup$ You can always define something. The question is: What do you want to achieve? $\endgroup$ – daw Nov 17 '17 at 9:53
  • $\begingroup$ So we can say $f(\cdot) \in \mathfrak{M}(\mathcal{X}; \mathcal{Y})$ instead of saying $f(\cdot)$ is measurable. So more information of $f(\cdot)$ can be shown by a single compact notation. $\endgroup$ – Johannes Nov 17 '17 at 10:00

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