I have a question about the notation used in this problem please dont solve the problem for me. Sho

Question

I have a question about the notation used in this problem please dont solve the problem for me.
Show that

| | X Y | |_{1} = | | X | |_{1} | | Y | |_{1}

for independent r.vs

X

and

Y

. Show further that if

X

and

Y

are also integrable, then

E (X Y) = E (X) E (Y)

I don't understand what the difference between

| | X | |_{1}

and

E (| X |)

. I assume we're talking about the

L^{1} (P)

norm? But then

| | X | |_{1} = \int_{Ω} | X (ω) | d P (ω) = E (| X |)

So if this is finite, then

E (X)

exists therefore it is integrable already (so why "also"?) Therefore the first "show that" would follow from the second

Answer 1

Under independence

| | X Y | |_{1} = | | X | |_{1} | | Y | |_{1}

holds even if

E | X |

or

E | Y |

is infinity (by Tonelli's Theorem). But

E (X Y) = (E X) (E Y)

holds if the expectations are finite.

Answers (1)