A function f(x) is said to be in Lp if the integral of |f(x)|p is finite. So if you know f is in Lp for some value of p, can conclude that f is also in Lq for some q? In general, no. But for two important special cases, yes. The special cases have to do with the domain of f.
If f is a function on a finite measure space and p > q, then every function in Lp is in Lq. In particular probability spaces are finite measure spaces, and so if a random variable’s pth moment exists, the qth moment exists.
If the domain of f is the integers with counting measure, f is a sequence and the “integral” of |f(x)|p is simply a sum. Now the relationship between Lp spaces is reversed: p > q, every sequence in Lq is in Lp.
There are two reasons a function could fail to be in Lp: singularities and fat tails. In a finite measure space, there are no tails; you only have to worry about singularities. On the integers, there are no singularities; you only have to worry about tails.
Now consider functions on the whole real line. Membership in Lp says nothing about membership in Lq because now you can have singularities and fat tails.