The random variable belongs to the class iff . The quantity is called the norm. The sequence converges in to iff .

(Convergence in Lp and in probability 1) If in then in probability.

By the formula ( Chebyshev inequality ) with we have and the claim follows.

(Convergence in Lp and in probability 2) If in probability and s.t. for all then in .

Fix a small , then The last expectation converges to 0 because converges to 0.