19.6.1.6 Lebesgue measure and integration

Problem: For the function \( f:[a, b] \rightarrow R \) a) find out if it's bounded; b) find the measure of the set of discontinuity points; c) find out if there is a proper or improper Riemann integral for it; d) find out if it's measurable \( f \); e) find the Lebesgue integral \( \int_{[a, b]} f(t) d t \), if it exists. \( a=-1, \quad b=1, \quad K \) is the Cantor set, \( f(t)=\left\{\begin{array}{cc}n, \quad t \in\left(\frac{1}{3^{n+1}}, \frac{1}{3^{n}}\right) \backslash K, n \in \mathbb{N}, \\ & {\left[e^{t^{2}}\right], \quad t \in K,} \\ \frac{1}{\sqrt{1+t}}, \quad & t \in\left([-1,0) \cup\left(\frac{1}{3}, 1\right)\right) \backslash K .\end{array}\right. \)