I am looking for a Lemma that gives an equivalent formulation for a family of functions to be equi-integrable: is it true that if $\{f_j\}_j\in L^1$, then we can write $f_j=f^1_j+f_j^2\in L^1+L^p$, where the norm of $f_j^1$ is arbitrarily small in $L^1$, and the $L^p$ norm of $f_j^2$ is fixed?
(Sorry, to be more specific: I know that this is true but I cannot find the correct reference.) Thanks in advance.
Given $\epsilon>0$, there are $N$, $M$ so that for all $j$,
$\ \ \ \ \ \ \int_{|x|\ge M} |f_j|<\epsilon$
and
$\ \ \ \ \ \ \int_{|f_j|>N} |f_j|<\epsilon$.
Let $A_j=\{\,x\mid |x|\ge M\,\}\cup \{ \,x\mid |f_j|>N\,\}$ and set $f_j^1=f_j\cdot\chi_{A_j}$.
The $L_p$-norm of the other component, $f_j-f_j^1$, will be bounded, since it's a bounded function with support contained in a set of finite measure.