  DEM analysis is a major diagnostic tool for stellar atmospheres. But both its
derivation and its interpretation are notably difficult because of random and
systematic errors, and the inverse nature of the problem. We use simulations
with simple thermal distributions to investigate the inversion properties of
SDO/AIA observations of the solar corona. This allows a systematic exploration
of the parameter space and using a statistical approach, the respective
probabilities of all the DEMs compatible with the uncertainties can be
computed. Following this methodology, several important properties of the DEM
inversion, including new limitations, can be derived and presented in a very
synthetic fashion.
  In this first paper, we describe the formalism and we focus on isothermal
plasmas, as building blocks to understand the more complex DEMs studied in the
second paper. The behavior of the inversion of AIA data being thus quantified,
and we provide new tools to properly interpret the DEM. We quantify the
improvement of the isothermal inversion with 6 AIA bands compared to previous
EUV imagers. The maximum temperature resolution of AIA is found to be 0.03 log
Te, and we derive a rigorous test to quantify the compatibility of observations
with the isothermal hypothesis. However we demonstrate limitations in the
ability of AIA alone to distinguish different physical conditions.
