About the paper from Marcinkiewicz, Jessen ad Zygmund

Question

I am studying the paper from Marcinkiewicz, Jessen ad Zygmund on differentiability of multiple integrals. At page 221 he writes \begin{equation*} f(P)=\phi(P)+\psi(P) \end{equation*} with \begin{equation*} \int_S|\psi(P)dP<\epsilon. \end{equation*} I guess this is possibile by density of continuous functions in $L^1$. Just below this, there is another equation that is \begin{equation*} \int_S|\psi(P)\log^+|\lambda\psi(P)|dP+\frac{B}{\lambda}<\epsilon. \end{equation*} I really don't get how is it possible to come to this inequality...since $\lambda$ is chosen large enough to have $\frac{B}{\lambda}<\frac{1}{2}\epsilon$, how can i bound a priori the integral also by $\frac{1}{2}\epsilon$?.