The Los Alamos free-electron laser (FEL) facility APEX is the first high-charge (greater than 1 nC), low-emittance (normalized rms emittance less than 5pi mm mrad) photoinjector-driven accelerator. The rapid beam acceleration, which is essential for maintaining high brightness, also prevents the beam from thermalizing its distribution. As a result, the phase-space the electron beam occupies appears as a contorted structure with a large surface area but little volume. The transverse beam shape is mostly non-Gaussian, often with a large halo at beam waists. This shape-often results in unusual effects, some of which are outlined in another paper for this conference. In this paper, we will examine how this physical property influences emittance measurement techniques, such as quadrupole scans and beam filtering through tubes and slits. We will show that common emittance measurement techniques typically underestimate the rms emittance by factors of two or more. We will also discuss the beam quality relevant to FEL interaction.