Two people can disagree about probabilities without disagreeing about what is
possible. A pessimist and an optimist may price the same coin differently, yet both
agree the coin can land heads and can land tails. When two
measures
\mathbb{P} and \mathbb{Q} agree on
exactly which events are impossible — the same null sets — we call them
equivalent and write \mathbb{P} \sim \mathbb{Q}.
Equivalence is the precise licence to translate one measure into the other. The dictionary is
a single non-negative random variable, the density (or
Radon–Nikodym derivative)
Z = \frac{d\mathbb{Q}}{d\mathbb{P}} \ge 0,
which reweights \mathbb{P} into
\mathbb{Q} outcome by outcome. Where Z > 1
the optimist counts an outcome as more likely than the pessimist; where
0 < Z < 1, less likely. Formally,
\mathbb{Q} measures a set A by summing
Z over it under \mathbb{P}:
\mathbb{Q}(A) = \int_A Z \, d\mathbb{P} \qquad \text{for every event } A.
From sets to expectations, derived line by line
The defining relation \mathbb{Q}(A) = \int_A Z\,d\mathbb{P} only
talks about probabilities of sets. We will upgrade it into the full
change-of-measure formula for
expectations,
\mathbb{E}_{\mathbb{Q}}[X] = \mathbb{E}_{\mathbb{P}}[XZ],
by the standard ladder of integration theory: start with indicators, climb to simple
functions, then to limits.
Step 1 — read the definition as an expectation. The integral of
Z over A is exactly the
\mathbb{P}-expectation of Z masked by the
indicator \mathbf{1}_A (which is 1 on
A and 0 off it):
\mathbb{Q}(A) = \int_A Z\,d\mathbb{P} = \mathbb{E}_{\mathbb{P}}[\mathbf{1}_A\, Z].
Step 2 — the indicator case of the formula. But the
\mathbb{Q}-probability of A is itself the
\mathbb{Q}-expectation of its indicator,
\mathbb{Q}(A) = \mathbb{E}_{\mathbb{Q}}[\mathbf{1}_A]. Comparing
with Step 1, the formula \mathbb{E}_{\mathbb{Q}}[X] = \mathbb{E}_{\mathbb{P}}[XZ]
already holds for every X = \mathbf{1}_A:
\mathbb{E}_{\mathbb{Q}}[\mathbf{1}_A] = \mathbb{E}_{\mathbb{P}}[\mathbf{1}_A\, Z].
Step 3 — extend to simple functions by linearity. A
simple random variable is a finite sum
X = \sum_{k} c_k\,\mathbf{1}_{A_k}. Both expectations are linear, so
applying Step 2 term by term,
\mathbb{E}_{\mathbb{Q}}[X] = \sum_k c_k\,\mathbb{E}_{\mathbb{Q}}[\mathbf{1}_{A_k}] = \sum_k c_k\,\mathbb{E}_{\mathbb{P}}[\mathbf{1}_{A_k} Z] = \mathbb{E}_{\mathbb{P}}\Big[\Big(\textstyle\sum_k c_k \mathbf{1}_{A_k}\Big) Z\Big] = \mathbb{E}_{\mathbb{P}}[XZ].
Step 4 — pass to the limit. Any non-negative measurable
X is an increasing limit of simple functions
X_n \uparrow X. Step 3 holds for each
X_n, and monotone convergence lets us take the limit inside both
expectations (note X_n Z \uparrow XZ since
Z \ge 0):
\mathbb{E}_{\mathbb{Q}}[X] = \lim_{n\to\infty} \mathbb{E}_{\mathbb{Q}}[X_n] = \lim_{n\to\infty} \mathbb{E}_{\mathbb{P}}[X_n Z] = \mathbb{E}_{\mathbb{P}}[XZ].
Splitting a general X = X^+ - X^- into positive and negative parts
extends it to any integrable X. The formula is proved.
The density integrates to one
Step 5 — take A = \Omega. The whole-space case of
the defining relation is the normalisation that pins Z down. Since
\mathbf{1}_\Omega = 1,
\mathbb{E}_{\mathbb{P}}[Z] = \int_\Omega Z\,d\mathbb{P} = \mathbb{Q}(\Omega) = 1,
because \mathbb{Q}, being a probability measure, assigns total mass
1. So a valid density is any non-negative
Z with \mathbb{P}-mean exactly
1: it is a reweighting that neither creates nor destroys total
probability. For the new measure to be equivalent (not merely absolutely continuous)
we need a touch more — Z > 0 almost surely — so that the dictionary
runs both ways and 1/Z = d\mathbb{P}/d\mathbb{Q}.
Let \mathbb{P} and \mathbb{Q} be
probability measures on (\Omega, \mathcal{F}). Then:
-
\mathbb{Q} is absolutely continuous with
respect to \mathbb{P} (written
\mathbb{Q} \ll \mathbb{P}, meaning
\mathbb{P}(A) = 0 \Rightarrow \mathbb{Q}(A) = 0) iff
there is a non-negative density
Z = d\mathbb{Q}/d\mathbb{P} with
\mathbb{Q}(A) = \int_A Z\,d\mathbb{P}. The two measures are
equivalent (\mathbb{P} \sim \mathbb{Q}) exactly
when Z > 0 almost surely.
-
The density is normalised:
\mathbb{E}_{\mathbb{P}}[Z] = 1.
-
Change of measure: for every integrable
X,
\mathbb{E}_{\mathbb{Q}}[X] = \mathbb{E}_{\mathbb{P}}[XZ].
The two notions are easy to confuse but do different jobs.
\mathbb{Q} \ll \mathbb{P} (absolute continuity) is one-directional:
anything \mathbb{P} rules out, \mathbb{Q}
rules out too — but \mathbb{Q} may have extra impossible
events of its own (those are the places where Z = 0).
Equivalence is the two-sided version,
\mathbb{Q} \ll \mathbb{P} and
\mathbb{P} \ll \mathbb{Q} together: the same null sets, no
exceptions, so Z > 0 everywhere and you can divide by it.
Statisticians know Z by another name: a
likelihood ratio. If \mathbb{P} and
\mathbb{Q} have densities p and
q against some common reference, then
Z(\omega) = \frac{d\mathbb{Q}}{d\mathbb{P}}(\omega) = \frac{q(\omega)}{p(\omega)},
the ratio of how plausible \omega is under the two hypotheses.
Every time you compute a likelihood ratio you are silently writing down a Radon–Nikodym
derivative.
This is the engine that the next pages run on. To build a risk-neutral world we will need a
density Z that converts the real-world measure into one in which
the discounted stock price is fair — and
Girsanov's theorem
will hand us exactly such a Z, built from the
exponential martingale.