The Conditioned Probability Measure Given a Set of Positive Measure

In the settings of The Real Numbers: Standing Notation and Background and Measure Spaces and the Lebesgue Integral: Standing Notation, the measure space of the latter being instantiated at each use by the measure space named, let $(S,\mathcal{S})$ be a measurable space, let $\nu$ be a probability measure on it, and let $A\in\mathcal{S}$ satisfy $0<\nu(A)$; then $\nu(A)\le\nu(S)=1$ by monotonicity of $\nu$, so $\nu(A)$ is a positive real number. For $B\in\mathcal{S}$ one has $B\cap A\in\mathcal{S}$ and $\nu(B\cap A)\le\nu(S)=1$ by monotonicity of $\nu$, so $\nu(B\cap A)/\nu(A)$ is a nonnegative real number. The function $h=\nu(A)^{-1}\mathbf{1}_{A}:S\to[0,\infty)$ is measurable by claims 1 and 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, the indicator function of a measurable set being measurable, and $\int_{S}\mathbf{1}_{B}\,h\,d\nu=\nu(A)^{-1}\nu(B\cap A)$ for $B\in\mathcal{S}$, because $\mathbf{1}_{B}h=\nu(A)^{-1}\mathbf{1}_{B\cap A}$.

(Conditioned probability measure)¶ The probability measure $\nu$ conditioned on $A$ is the function

$$\nu(\cdot\,|\,A):\mathcal{S}\to[0,\infty),\qquad\nu(B\,|\,A)=\frac{\nu(B\cap A)}{\nu(A)} .$$

It is the measure with density $h$ with respect to $\nu$ of claim 3 of Image Measures, Measures with Densities, and Change of Variables, by the identity above, and it is a probability measure, since $\nu(S\,|\,A)=\nu(A)/\nu(A)=1$.