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$\sigma-$ -> $\sigma$-
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LSpice
LSpice
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I recently learned the mathematical definition of a random variable, namely:

A random variable is a measurable function $X: \Omega \rightarrow \mathbb{R}$ whose domain $\Omega$ is equipped with a $\sigma-$$\sigma$-algebra $\mathcal{F}$ and a probability measure $P$.

This definition seems unintuitive to me, so I wanted to see how some of the famous classes of random variables are defined using the definition (e.g. Gaussian, Uniform). However, from what I understand in these cases there is no clear sample space $\Omega$ but rather a pdf which is used to define the function. That got me thinking why can't we just define a random variable as a probability measure on the Borel sets instead of using another measure on another $\sigma-$$\sigma$-algebra and then requiring the function to be measurable?

I recently learned the mathematical definition of a random variable, namely:

A random variable is a measurable function $X: \Omega \rightarrow \mathbb{R}$ whose domain $\Omega$ is equipped with a $\sigma-$algebra $\mathcal{F}$ and a probability measure $P$.

This definition seems unintuitive to me, so I wanted to see how some of the famous classes of random variables are defined using the definition (e.g. Gaussian, Uniform). However, from what I understand in these cases there is no clear sample space $\Omega$ but rather a pdf which is used to define the function. That got me thinking why can't we just define a random variable as a probability measure on the Borel sets instead of using another measure on another $\sigma-$algebra and then requiring the function to be measurable?

I recently learned the mathematical definition of a random variable, namely:

A random variable is a measurable function $X: \Omega \rightarrow \mathbb{R}$ whose domain $\Omega$ is equipped with a $\sigma$-algebra $\mathcal{F}$ and a probability measure $P$.

This definition seems unintuitive to me, so I wanted to see how some of the famous classes of random variables are defined using the definition (e.g. Gaussian, Uniform). However, from what I understand in these cases there is no clear sample space $\Omega$ but rather a pdf which is used to define the function. That got me thinking why can't we just define a random variable as a probability measure on the Borel sets instead of using another measure on another $\sigma$-algebra and then requiring the function to be measurable?

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Syail
Syail
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Why do we need to define a random variable as a function?

I recently learned the mathematical definition of a random variable, namely:

A random variable is a measurable function $X: \Omega \rightarrow \mathbb{R}$ whose domain $\Omega$ is equipped with a $\sigma-$algebra $\mathcal{F}$ and a probability measure $P$.

This definition seems unintuitive to me, so I wanted to see how some of the famous classes of random variables are defined using the definition (e.g. Gaussian, Uniform). However, from what I understand in these cases there is no clear sample space $\Omega$ but rather a pdf which is used to define the function. That got me thinking why can't we just define a random variable as a probability measure on the Borel sets instead of using another measure on another $\sigma-$algebra and then requiring the function to be measurable?