I was trying to solve the question but i wasn't sure how to attempt

for each p>0, let L^p denote the collection of all such random variables on R^k for which the pth moment is finite: E|X|^p <∞

i. L^p is a vector space

I don't know how to prove it...

ii. for any 0<r<p, E|X|^p <∞ => E|X|^r <∞

I tried to prove this...

From the simple inequality a^r ≤ 1+a^p, all a ≥ 0 and 0 < r < p

now if i let a = |X| and tale expectations, I obtain E|X|^r ≤ 1+E|X|^p

So, for p>0, the pth moment of x is finite, then also the rth moment is finite for any 0<r<p

iii. E|X|^p <∞ iff E|Xi|^p <∞, i=1,...,k

iff E|t'X|^p <∞, for all t in R^k

I thought I need to use i and ii to solve iii... and for part iii, so I tried to prove this...

|Xi| ≤ |X| ≤ ∑ |Xi|

L^p = {X in R^k: E|X|^p <∞} is a linear space

since for any 0<r<p, E|X|^p <∞ => E|X|^r <∞, i dont know...

Someone plz help me???