  In this paper, we show how the Robertson uncertainty relation gives certain
intrinsic quantum limits of measurements in the most general and rigorous
mathematical treatment. A general lower bound for the product of the
root-mean-square measurement errors arising in joint measurements of
noncommuting observables is established. We give a rigorous condition for
holding of the standard quantum limit (SQL) for repeated measurements, and
prove that if a measuring instrument has no larger root-mean-square
preparational error than the root-mean-square measurement errors then it obeys
the SQL. As shown previously, we can even construct many linear models of
position measurement which circumvent this condition for the SQL.
