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{\bf Question}

Calculate the Lebesgue integral $\ds L\int_{[0,1]}f$ where $f:{\bf
R}\rightarrow{\bf R}$.

$f(x)=\left\{\begin{array}{cl} 1&amp; x {\rm \ is \ rational}\\ 0&amp; x
{\rm \ is \ irrational}\end{array}\right.$


\vspace{0.25in}

{\bf Answer}

${}$

$\ds L\int_{[0,1]}f \hspace{0.5in}$  Let $g=0, \,\,\, f=g$ a.e.

So $\ds L\int_{[0,1]}f=L\int_{[0,1]}g=R\int_{[0,1]}g=0$


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