Change of measure and density martingales

Let Z be a nonnegative integrable random variable of expectation one under a base probability P. Defining Q of an event as the P-expectation of Z times its indicator gives a probability measure absolutely continuous with respect to P. If Z is strictly positive P-almost surely, the two measures are equivalent, meaning they have the same null events. Nonnegativity alone gives only one direction of absolute continuity; events supported where the density vanishes can have positive P-probability and zero Q-probability.

For an increasing filtration define the density process by conditional expectation of Z. The tower property makes it a nonnegative P-martingale. On each time-n event, integrating against this conditional density agrees with integrating against Z, so the process is exactly the Radon–Nikodym density of the restricted measure at that time. Because one integrable variable closes the process, it is uniformly integrable. Its limit is the conditional expectation of Z given the eventually observed sigma-algebra, equal to Z when that sigma-algebra contains all its information.

For a Q-integrable variable X, the product ZX is P-integrable, and conditional Bayes says its P-conditional expectation divided by the current density is the Q-conditional expectation of X. The formula holds on the set where that density is positive. Its complement is Q-null, so arbitrary values can be assigned there for a Q-version; insisting on a ratio everywhere would create an unnecessary division by zero. Strictly positive terminal density ensures positive current densities P-almost surely and avoids that ambiguity.

If X is measurable at a later filtration time m, the numerator may use the time-m density in place of the terminal density, by conditioning first at m. The ratio still divides by the density at the earlier conditioning time n. This order is crucial: the density process is a P-martingale, but a P-martingale need not remain a Q-martingale after the law changes. Conditional means must be recalculated under the new weights. A finite coin model makes every term in the formula directly verifiable.

Conversely, a nonnegative mean-one martingale supplies a consistent density on each finite-time sigma-algebra. Without uniform integrability its almost-sure limit can have mean less than one, preventing that limit from serving as a probability density with the original normalization. Additional measure-extension arguments may still construct a law on the full sequence space, but that law need not be absolutely continuous with respect to P on the eventual sigma-algebra. Finite-horizon consistency and terminal absolute continuity are different assertions.

For independent fair tosses under P and independent tosses with heads probability three quarters under Q, every finite likelihood ratio is strictly positive. Yet the strong law gives different limiting heads frequencies under the two infinite laws, making them singular on the full sequence space. Under P the likelihood ratio's logarithm has a negative linear limiting slope, so the ratio tends to zero almost surely despite expectation one at every finite time. Positivity at each finite horizon therefore cannot replace uniform integrability when claiming one global terminal density.