  We demonstrate high fidelity repetitive projective measurements of nuclear
spin qubits in an array of neutral ytterbium-171 ($^{171}$Yb) atoms. We show
that the qubit state can be measured with a fidelity of 0.995(4) under a
condition that leaves it in the state corresponding to the measurement outcome
with a probability of 0.993(6) for a single tweezer and 0.981(4) averaged over
the array. This is accomplished by near-perfect cyclicity of one of the nuclear
spin qubit states with an optically excited state under a magnetic field of
$B=58$ G, resulting in a bright/dark contrast of $\approx10^5$ during
fluorescence readout. The performance improves further as $\sim1/B^2$. The
state-averaged readout survival of 0.98(1) is limited by off-resonant
scattering to dark states and can be addressed via post-selection by measuring
the atom number at the end of the circuit, or during the circuit by performing
a measurement of both qubit states. We combine projective measurements with
high-fidelity rotations of the nuclear spin qubit via an AC magnetic field to
explore several paradigmatic scenarios, including the non-commutivity of
measurements in orthogonal bases, and the quantum Zeno mechanism in which
measurements "freeze" coherent evolution. Finally, we employ real-time
feedforward to repetitively deterministically prepare the qubit in the $+z$ or
$-z$ direction after initializing it in an orthogonal basis and performing a
projective measurement in the $z$-basis. These capabilities constitute an
important step towards adaptive quantum circuits with atom arrays, such as in
measurement-based quantum computation, fast many-body state preparation,
holographic dynamics simulations, and quantum error correction.
