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I saw that in is sometimes used and so is on.

For example,

Let $f$ and $f_k$, $k=1, 2, \cdots$, be measurable and finite a.e. in $E$. If $f_k\to{}f$ a.e. on $E$ and $|E|\lt+\infty$, then $\{f_k\}$ converge in measure on $E$ to $f$.

In the theorem above, both in $E$ and on $E$ are used. I am wondering the difference between them. For your information, a.e. means almost everywhere.

Asaf Karagila
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Danny_Kim
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1 Answers1

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in means that something is a member of a se, e.g. $2$ is in $\mathbb N$

on means that something is true for all elements of a set, e.g. $|x|$ is continuous on $\mathbb R\setminus\{0\}$.

Henrik supports the community
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  • $E$ is a set of $\mathbf{x}$, and $f$ is a function of the $\mathbf{x}$. That is, $f$ is not a member of $E$. However, my example (from the textbook) writes $f$ is measurable in $E$ and $f$ is finite a.e. in $E$. Why does the author use in here? – Danny_Kim May 08 '16 at 10:02
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    The a.e. makes the difference here, it means that we go from talking about $f$ to talking about points in the domain (which we then restrict to $E$). – Henrik supports the community May 08 '16 at 10:18
  • Ah!! $f$ and $f_k$ is measurable and finite almost every $\mathbf{x}$ in E. Thank you – Danny_Kim May 08 '16 at 10:27