I have a question about the following reasoning: 

Let $(X,\mathcal{M}, \mu)$ be a measure space
and assume that $\mu(X)$ is finite. 

Then by Holder, we know that there exists a constant $C\_{p,q}$
such that $$||f||\_p \leq C\_{p,q} ||f||\_q$$, when $p<q$.

Now, consider the identity map $i\colon L^p(\mu)\rightarrow L^q(\mu)$,
Let $(f\_i,f\_i)\in Graph(i)$, be a sequence in the product space converging to $(f,g)$, which means that $f\_i$ converge to $f$ in $L^p$
and $f\_i$ converge to $g$ in $L^q$. Then we have that $||f\_i-g||\_p \leq C\_{p,q} ||f\_i-g||\_q$, which means that $f\_i$ converge to $g$ in $L^p$, then $f=g$ almost everywhere. Since the identity map is continuous (by closed Graph Theorem)
we conclude that there exists a constant $C'\_{p,q}$ such that
$||f||\_q \leq C'\_{p,q}||f||\_p$. Thus, the space $X$ does not contain elements of arbitrarily small measure... But that is not true in general.

Can someone tell me what I am doing wrong?