Difference between revisions of "Measurable map"

Revision as of 23:03, 2 May 2015

Definition

Let $(X,\mathcal{A})$ and $(X',\mathcal{A}')$ be measurable spaces

Then a map $$T:X\rightarrow X'$$ is called $$\mathcal{A}/\mathcal{A}'$$ -measurable if

$$T^{-1}(A')\in\mathcal{A},\ \forall A'\in\mathcal{A}'$$

Notation

A given a measure space (a measurable space equipped with a measure) $(X,\mathcal{A},\mu)$ with a measurable map on the following mean the same thing:

• $$T:(X,\mathcal{A},\mu)\rightarrow(X',\mathcal{A}',\bar{\mu})$$ (if $(X',\mathcal{A}')$ is also equipped with a measure)
• $$T:(X,\mathcal{A},\mu)\rightarrow(X',\mathcal{A}')$$
• $$T:(X,\mathcal{A})\rightarrow(X',\mathcal{A}')$$

We would write $$T:(X,\mathcal{A},\mu)\rightarrow(X',\mathcal{A}')$$ simply to remind ourselves of the measure we are using, it is not important to the concept of the measurable map.

Motivation

From the topic of random variables - which a special case of measurable maps (where the domain can be equipped with a probability measure, a measure where $X$ has measure 1).

Consider: $$X:(\Omega,\mathcal{A},\mathbb{P})\rightarrow(V,\mathcal{U})$$ , we know that given a $U\in\mathcal{U}$ that $T^{-1}\in\mathcal{A}$ which means we can measure it using $\mathbb{P}$, which is something we'd want to do.

See also

• Random variable
• Probability space
• Measurable space

References