# Measurable variables

Consider $$\pi$$ that is measurable variable with respect to the $$\sigma-\text{algebra}$$ $$\mathcal{F}$$. Is this equivalent to write $$\pi\in\mathcal{F}$$? By using the concept of measurability it means that $$\pi$$ as measurable is known and we can find it or simply we can use is as a constant am I right?

## 1 Answer

A function $$\pi:(\Omega_1,\mathcal{F}_1)\to(\Omega_2,\mathcal{F}_2)$$ is said to be measurable if given $$A\in\mathcal{F}_2$$ ($$A$$ is a set of that belongs to the $$\mathcal{F}_2$$ $$\sigma$$-algebra), then $$\pi^{-1}(A)\in\mathcal{F}_1$$.

Some books use the notation $$\pi\in\mathcal{F_1}$$ as a shorthand for "$$\pi$$ is measurable with respect to $$\mathcal{F}_1$$". But, obviously, $$\pi$$ is not a subset of $$\Omega_1$$, so it can't be an element of $$\mathcal{F}_1$$. It's just a kind of abuse of notation that should be unambiguous and harmless.

• well thank you very much. It is not some books, it is in the vast majority of every graduate and undergraduate book... – Hunger Learn Jun 7 '20 at 9:41