How we should calculate the Lebesgue decomposition of a measure? Please explain it with an example such I can get the whole idea behind it.
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Let e.g. $\nu$ be a finite Borel measure on $[a,b]$ and $m$ the Lebesgue measure. So the function $[a,b]\ni x\mapsto \nu\big([a,x)\big)$ is a BV function. It is therefore differentiable $m$a.e., and its derivative $\rho(x)$ coincides with the RadonNikodym derivative of the absolutely continuous part of $\nu$. Knowing $\rho$ you can deduce the Lebesgue decomposition of $\nu$ wrto the Lebesgue measure: $\nu=\nu _ a+\nu _ s$ with $\nu _a\perp\nu _ s$ where $\nu_a(E)=\int_E\rho(x)dm(x)$. 

