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Math Help - countable set

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    countable set

    is there any countable set which is not measurable? if there is, please help me.
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    Quote Originally Posted by Kat-M View Post
    is there any countable set which is not measurable? if there is, please help me.
    With respect to a particular measure?

    For the Lebesgue measure the answer is no, since any countable set is measurable with measure 0.
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    Quote Originally Posted by pomp View Post
    With respect to a particular measure?

    For the Lebesgue measure the answer is no, since any countable set is measurable with measure 0.
    with respect to any measure. so for a different measure, there is a countable set which is not measurable?
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    Quote Originally Posted by Kat-M View Post
    with respect to any measure. so for a different measure, there is a countable set which is not measurable?
    My intuition says no, it's my understanding that any countable set has measure zero meaning it must be measurable, but the only measure I have ever worked with is the Borel and Lebesgue measures. Measure Theory is not really my thing, and it's also 7:48 and I haven't slept yet I could be wrong.

    I'm sorry, but I'm sure your question will be answered very swiftly by someone more confident than me!
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    Hello,

    There are many things you have to define before talking about measurability !

    A counterexample is :

    Let's deal with (\mathbb{R},\mathcal{A},\lambda), where \mathcal{A}=\{\mathbb{R},\emptyset,\{0\},\mathbb{R  }\backslash \{0\}\} and \lambda is the Lebesgue measure.
    The only measurable sets are the ones in \mathcal{A}.
    Hence \mathbb{N}, which is a countable set, is not measurable !
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