Final answer:
An investigator would use an EEG (Electroencephalogram) to identify a person's sleep stages because it measures the electrical activity of the brain, capturing unique wave patterns associated with each stage.
Step-by-step explanation:
An investigator wishing to identify whether a person is in stage 1, 2, 3, or 4 of sleep would most likely rely on an EEG (Electroencephalogram). The EEG is a test that measures the electrical activity of the brain and is a useful tool for monitoring sleep stages because it can capture the brain wave patterns unique to each stage of sleep. Other imaging techniques, such as MRI (Magnetic Resonance Imaging), PET (Positron Emission Tomography), and CT (Computed Tomography) scans, can provide images of the brain's structure or metabolic activity, but do not offer the temporal resolution necessary to differentiate between the various stages of sleep.