  Indeterminacy associated with probing of a quantum state is commonly
expressed through spectral distances (metric) featured in the outcomes of
repeated experiments. Here we express it as an effective amount (measure) of
distinct outcomes instead. The resulting $\mu$-uncertainties are described by
the effective number theory [1] whose central result, the existence of a
minimal amount, leads to a well-defined notion of intrinsic irremovable
uncertainty. We derive $\mu$-uncertainty formulas for arbitrary set of
commuting operators, including the cases with continuous spectra. The
associated entropy-like characteristics, the $\mu$-entropies, convey how many
degrees of freedom are effectively involved in a given measurement process. In
order to construct quantum $\mu$-entropies, we are led to quantum effective
numbers designed to count independent, mutually orthogonal states effectively
comprising a density matrix. This concept is basis-independent and leads to a
measure-based characterization of entanglement.
