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Let $(X,\mathcal{M},\mu)$ be a measure space and let $E\in \mathcal{M}$. I'm interested in knowing what the default way to define a measure space $(E,\mathcal{M}_E,\mu_E)$ is?

My guess would be the $\sigma$-algebra $\mathcal{M}_E = \{F \cap E : F\in \mathcal{M} \}$ defined $E$ with measure $\mu_E = \mu|_{\mathcal{M}_E}$. In lieu of other information, is this the correct measure space to assume? Also, is there any other sensible way to define it?

I'm asking because I know what it means for $f:\mathbb{R} \rightarrow \mathbb{R}$ to be Lebesgue or Borel measurable, but want to make sure I'm not assuming the wrong thing for when I see someone say that $f:[0,1] \rightarrow \mathbb{R}$ is Lebesgue or Borel measurable.

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    $\begingroup$ Talking again about alternative definitions, sometimes people write $\mathcal M_E = \{F:F\in \mathcal M, F\subseteq E\}$ - which however coincides with the definition you provided. $\endgroup$
    – SBF
    Commented Jun 1, 2013 at 12:11

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Your guess is right; it's just a restriction of the measure on the larger space.

On a related note, in probability theory one works in a space whose total measure is $1$ and one often considers the measure $$ \frac{\mu|_{\mathcal{M}_E}}{\mu(E)}. $$ Values of this measure are "conditional probabilities given the event $E$".

Later postscript: Actually conditional probabilities given $E$ are often assigned to sets that are not subsets of $E$. The conditional probability of $A$ given $E$ is the value assigned to $A\cap E$ by the measure given in my second paragraph.

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