Question

Show using an example 𝑓 and 𝑔 are not integrable on [𝑎, 𝑏], but 𝑓𝑔 may be integrable on [𝑎, 𝑏].

Answer 1

f(x)=$\{\begin{matrix} 1,if x\epsilon [0,1]\\ 0,elsewhere \end{matrix}$

g(x)=$\{\begin{matrix} 0,if x\epsilon [0,1]\\ 1,elsewhere \end{matrix}$

f.g=0

Here, f and g are not integrable but f.g is integrable( since constant function).