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New comer to measure theory! I'm struggling to wrap my head around this concept...

Referring to Lebesgue Measures!

Could someone provide a concrete example of a set in the interval $[0,1]$ such that the probability of choosing a point in that set is not well defined?

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    Do you mean find a subset of $[0,1]$ such that its points are not Lebesgue measurable? Or do you mean find $\sigma$-algebra on $[0,1]$ such that (some) points are not measurable? – 2017-01-23
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    It sounds like you are looking for a subset of $[0,1]$ that is not Lebesgue measurable. If so, the link is a duplicate. If not, please explain the difference and we can work on it. – 2017-01-23
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    @Ennar Ah i mean to say a subset of [0,1] such that its points are not Lebesgue measurable – 2017-01-23
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    I agree with Ross, you are probably looking for non-measurable set. Note that $\{x\}$ is always Lebesgue measurable with measure $0$, so "its points are not Lebesgue measurable" is probably not what you really want. – 2017-01-23

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