= Solution
A <weak measurement> couples an <observable> only weakly to a meter, so the pointer distributions associated with its different eigenvalues overlap strongly. One trial supplies little information and a small disturbance; an ensemble is needed to estimate a small mean shift. It is not an exact determination of a previously hidden eigenvalue with no disturbance.
For pointer position $q$ and conjugate momentum $p$, a short <von Neumann measurement interaction> has unitary $U=\exp(-igA\otimes p/\hbar)$. An eigenvalue $a$ translates the pointer by $ga$. The weak regime requires those translations to be small relative to the pointer width. Without postselection the first-order mean displacement is $g\langle A\rangle$ for an initially uncorrelated, suitably centered meter.
If the system is prepared in $|i\rangle$ and postselected in $|f\rangle$, the unnormalized meter state is
$$
\langle f|U|i\rangle|\varphi\rangle\simeq\langle f|i\rangle\left(1-\frac{ig}{\hbar}A_w p\right)|\varphi\rangle,\qquad
\boxed{A_w=\frac{\langle f|A|i\rangle}{\langle f|i\rangle}.}
$$
For a real centered Gaussian pointer with no position–momentum correlation, the conditional position shift is $g\operatorname{Re}A_w$; the imaginary part affects the conjugate momentum distribution. The <weak value> can be complex or have a real part outside the observable's eigenvalue interval, because it is a conditional transition-amplitude ratio, not an eigenvalue.
An example apparatus uses photon polarization as the two-level system, a weak polarization-dependent transverse displacement as the interaction, and an output polarizer for <postselection>. Repeated position detections estimate the selected beam's displacement. Nearly orthogonal selection can make the conditional shift large, but also reduces the accepted sample and eventually invalidates the first-order expansion. The original weak-value proposal emphasized preselection, a weak coupling and postselection, rather than an anomalous result from an ordinary sharp spin measurement.<Weak values by Yakir Aharonov, David Z. Albert and Lev Vaidman>[1988 primary paper: preselected and postselected spin ensemble and conditional weak value]{ref}
Weak coupling limits <quantum measurement backaction> per trial; it does not abolish the information–disturbance tradeoff. Repeating weak observations can accumulate substantial backaction, and discarding rejected trials cannot automatically improve the information available per prepared system. Conditional pointer amplification and a statistically improved parameter estimate are distinct claims.
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