Really basic question here since I'm a beginner.

Obviously the Lebesgue measure on $\mathbb{R}^n$ isn't a probability measure (since $\mu(\mathbb{R}^n) \neq 1$), but what if we have the Lebesgue measure restricted to a set of measure 1 in $\mathbb{R}^n$? That is, say we have the box $B = [0,1]^n$. Then is the Lebesgue measure restricted to subsets of $B$ a probability measure, or is there something else I'm missing? If so, would it be considered the "canonical" probability measure here?

  • 2
    $\begingroup$ If $A$ has measure one, then $\mu B = m(B \cap A)$ is a probability measure on the real line. (This is true for any underlying measure, not just the Lebesgue measure). $\endgroup$ – copper.hat Feb 19 '18 at 21:54

Yes it will be a probability measure. In fact it is the uniform measure on $B $ (uniform distribution: https://en.m.wikipedia.org/wiki/Uniform_distribution_(continuous))


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.