Preparation region

In plain terms

Here is the problem. No laboratory procedure controls a system completely. Cool an atom, align a polariser, fire the same laser -- what you fix is everything your instruments can see, and the world keeps the rest. So "prepare the same state a thousand times" cannot mean "the same world-state a thousand times".

The preparation region is the honest version: the set of all arena points compatible with what the procedure fixed. Each run of the experiment is one point drawn from that region. The statistics of many runs are the statistics of the region.

Once you say it this way, quantum probability stops being mysterious in kind: it is the same ignorance-over-a-region that classical statistical mechanics runs on. The work is in proving the numbers come out right, and that is where the measure theorems earn their keep.

In CSD

The preparation region is a primitive of the LF1 layer: a measurable subset of Sigma with positive Liouville volume, from which repeated trials draw independently under the conditioned measure. Every frequency theorem in the corpus is stated relative to it -- the outcome weight is the volume of the outcome's preimage intersected with the preparation region, normalised by the region's volume.

The pure-state refinement has its own honesty note: under the continuous measure bridge a single state's fibre is a null set, so pure preparations carry a posited fibre trial law rather than a conditional of the ambient measure -- stated as such where it is used.

Mathematically

prepMeasure conditions the Liouville measure on the preparation region; repeated trials are i.i.d. draws from it; the LF1 main theorem sends empirical frequencies to volume weights almost surely. The conditional-measure formula and the weight-as-volume-ratio identity are the exported infrastructure the higher layers consume.