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I'm having difficult time in understanding the difference between the Borel measure and Lebesgue measure. Which are the exact differences? Can anyone explain this using an example?

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Not every subset of a set of Borel measure $0$ is Borel measurable. Lebesgue measure is obtained by first enlarging the $\sigma$-algebra of Borel sets to include all subsets of set of Borel measure $0$ (that of courses forces adding more sets, but the smallest $\sigma$-algebra containing the Borel $\sigma$-algebra and all mentioned subsets is quite easily described directly (exercise if you like)).

Now, on that bigger $\sigma$-algebra one can (exercise again) quite easily show that $\mu$ (Borel measure) extends uniquely. This extension is Lebesgue measure.

All of this is a special case of what is called completing a measure, so that Lebesgue measure is the completion of Borel measure. The details are just as simple as for the special case.

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Not every subset of a set of Borel measure $0$ is Borel measurable. It is important specify this. –  leo Dec 31 '12 at 20:23
    
thank you leo, I updated the post. –  Ittay Weiss Dec 31 '12 at 20:40
    
You are welcome –  leo Dec 31 '12 at 20:46

borel measure is defined on the smallest sigma algebra that contains all the open sets while lebesgue measure is much much more general but coincides with borel,when a set is a borel measurable.there are examples (not so easy) of non-borel lebesgue measurable sets

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