  In order to bypass the local measurement as an intermediate quantity in the
proof of the eigenstate thermalization hypothesis (ETH), we have proposed a
measurement-independent distinguishability measure. In this paper, we establish
the connection between this measure and several other ETH measures in a more
unified way. We first prove a universal trade-off relation for the diagonal and
off-diagonal elements of the measure. Then we extend it to the discussion of
eigenstate typicality and the average observable. In the trade-off
relationship, the exponential increase of off-diagonal elements will directly
suppress the value of off-diagonal elements, and the diagonal elements will
also be suppressed indirectly. This relationship gives another perspective to
understand the physical mechanism behind ETH. Finally, by numerically
calculating one-dimensional Ising spin chain, we examine several trade-off
relationships and strong and weak ETH.
