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?
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.