5) For A C R, the outer measure of A is defined by
(A=inf{:AU f=1
where I is an open interval and if I=(,b,, then |I|=b, -, j =1,2,..-
Let{} sequence of real numbers and A={ :n N} C Io=[ab],where abRandab.
1,
Using the definition of * given above, show that *(A) = 0.
ii, Prove that A is Lebesgue measurable by showing that
(A)= (A), where A)= Io|=* (IA) is the inner measure of A,
and find Lebesgue measure (A) of A. iji. Argue that IA is Lebesgue measurable and show that
IoA=|I|=b-a
Recall, h : [,b] R is measurable if for all R, h-1([,)) is Lebesgue measurable (i.e., h([,)) 1, where 1 is the set of all measurable subsets of [a, b]). i, Let f : [0,] R be given by
1xE[0,]Q fx= 0x[0,]nQ
Using the definition given above, show that g : [0, ] R defined by
gx=max{sinxfx}
is measurable
ii, Using properties of Lebesgue integral, find the Lebesgue integral Jjo, () d
