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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?

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    $\begingroup$ 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
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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))

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