Discuss the below:
Q1 Prove directly from Proposition 17.2 that each of the following function is measurable on [0,1].
a f(x)=3 for all x∈[0,1]
b f(x)=x for all x∈[0,1]
c f(x)={1/x if x∈(0,1]
0 ifx=0}
Q2 Proposition: Let A be bounded and measurable and . Then the following are equivalent:
1. {x∈A|c
2. {x∈A|c<=f(x)} is measurable for all c
3. {x∈A|f(x)
4. {x∈A|f(x)<=c} is measurable for all c
5. {x∈ A|c
6. {x∈A|c<=f(x)
7. {x∈A|c<=f(x)<=d} is measurable for all c < d
8. {x∈ A|c
In other words, any of the conditions 1 - 8 could be taken as the definition of measurability.