What are you looking for? For #1, use the definition of a measurable function. For #2, use the definition of an integrable function.
Let f : (0,1) —>R be measurable( w.r.t. Lebesgue measure) function in L1((0,1)). Define the function g on (0,1)× (0,1) by
g(x,y)=f(x)/x if 0<y<x<1
g(x,y)=0 if 0<x≤y<1
Prove:
1) g is measurable function (w.r.t. Lebesgue measure in the prodcut (0,1)× (0,1)
2)g is integrable in (0,1)× (0,1)