I'm currently reading this pdf: https://uu.diva-portal.org/smash/get/diva2:1231351/FULLTEXT01.pdf On Singular Integral Operators, Marcus Vaktnäs. 
In the fourth line it says "so at least one of $\langle |f| \rangle _I$ and $\langle |f| \rangle _J$ are still greater than $\lambda$". This may be very dumb but I don't see how that is true? In the previous line it shows that the sum is smaller than $2\lambda$, couldn't both of them be smaller than $\lambda$?