Differences between the Borel measure and Lebesgue measure

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?

Answer

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

Now, on that bigger σ-algebra one can (exercise again) quite easily show that μ (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.

Attribution
Source : Link , Question Author : Mark Hyatt , Answer Author : Ittay Weiss

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