Suppose we have a sequence of random variables, $$ (Z_p)$$ with $$\mathbb{E}Z_p=1,\qquad Z_p\ge 0 \quad a.s. $$ and $$ Z_p\to 1 \quad \text{in probability}$$
We can chose a subsequence, $Z_{p_i}$ such that $$\mathbb{P}(|Z_{p_i}-1|>1/i)<1/i $$ then $$Z_{p_i}\to 1 \quad a.s.$$ By Sheffe's lemma, we have $$\mathbb{E}|Z_{p_i}-1|\to 0 $$.
The question now is, can we conclude that$$\mathbb{E}|Z_{p}-1|\to 0 $$