Please help me with this problem needed all parts and clearly explain please

Problem 1. Recall that we constructed the Lebesgue measure on [0, 1]. Call it /jo,1). Notice that we could've done that for any interval [a, b]. 1. [Not to be turned in] Check for yourself that " = >-_Aqui+1) defines a o-finite measure on R. This is called the Lebesgue measure on R. 2. Show that this measure satisfies Fatou's lemma using Fatou's lemma for probability measures. 3. [Not to be turned in] Check for yourself that Fatou, MCT, and DCT for general measure spaces with o-finite measures using the same idea