19.6.1.9 Lebesgue measure and integration

Problem: For the function \( f \), given on the segment \( [a, b] \) 1) prove that it's simple; 2) calculate the Lebesgue integral \( \int_{[a, b]} f(t) d t \), if it does exist. \[ a=0, \quad b=1, \quad f(t)=\left[\frac{1}{\sqrt{t}}\right] . \]