Two measures assign different weights to the same atoms. If μ gives an atom mass two and ν gives it mass six, the relative density must be three there. If μ gives another atom mass zero but ν gives it mass one, no density can repair the mismatch: multiplying a μ-null set by any density still gives zero integral. This null-set condition is absolute continuity.
Predict
Predict whether absolute continuity refers to closeness of total masses. It does not. It means μ(E)=0 implies ν(E)=0 for every measurable E. A density may be very large or unbounded while satisfying this condition. Predict also whether densities are unique at individual null points; only their almost-everywhere equivalence class is determined.
Experiment
On a finite three-atom space, calculate relative densities.
Atom
μ mass
ν mass
Density
a
1
3
3
b
2
1
1/2
c
3
0
0
Verify the measure of each union by summing density times μ mass. Then change only the μ mass of a to zero. Identify exactly which hypothesis fails and why no choice of a finite or infinite density fixes it.
Understand
The Radon–Nikodym theorem for sigma-finite positive measures states that ν≪μ implies ν(E)=integral_E f dμ for a nonnegative measurable f, unique μ-almost everywhere. We prove the finite case and reduce the sigma-finite case to it.
Set λ=μ+ν. On the real Hilbert space L2(λ), the functional T(h)=integral h dν is bounded: Cauchy–Schwarz for ν bounds it by sqrt (ν(X)) times norm (h) in L2(λ). The Hilbert representation theorem gives g with T(h)=integral hg dλ. Here is the mechanism behind that theorem. For a nonzero bounded functional, minimize norm on the closed affine set T(h)=1. The parallelogram identity makes a minimizing sequence Cauchy; completeness gives a minimizer u. Varying u along the kernel proves u is orthogonal to that kernel. Decomposing any h into T(h)u plus a kernel vector gives T(h)=inner (h, u)/norm (u)². The zero functional is immediate.
Testing indicator functions gives ν(E)=integral_E g dλ. Since ν is positive, threshold-set tests imply g≥0 almost everywhere. Also μ(E)=integral_E(1−g)dλ, so g≤1 almost everywhere. On the set where g=1, μ is zero; ν≪μ forces ν and hence λ to be zero there. Define f=g/(1−g) off that null set, and zero on it. The simple-function integration identity for μ with density 1−g relative to λ gives integral_E f dμ=integral_E g dλ=ν(E).
Uniqueness follows by testing sets where one candidate exceeds another by at least 1/k. For sigma-finite μ and ν, partition X into countably many disjoint measurable pieces on which both measures are finite, obtained from intersections of their finite-measure exhaustions. Apply the finite result on each piece and paste the densities. Nonnegative summation proves the global representation.
Apply
Check the quotient construction against the atomic table: g equals ν mass divided by the combined mass, so g/(1−g) equals ν mass divided by μ mass. The abstract Hilbert argument reconstructs the same ratio without requiring individual atoms. For signed finite measures, apply the theorem to the positive and negative Jordan parts when their variation is absolutely continuous.
Continue
A density is always relative to a reference measure. The next lesson separates the part of a measure that has such a density from a singular part concentrated on a reference-null set. This distinction will later explain both conditional expectation and singular probability laws.
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On an atom, μ has mass 4 and ν has mass 10. What value must dν/dμ have there?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
On [0,1], ν has density 6x relative to length. What is ν([0,1/2])?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Prove uniqueness almost everywhere of an integrable density representing a finite positive measure.
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Show that a density representation ν(E)=integral_E f dμ implies ν is absolutely continuous with respect to μ.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
In the finite-measure proof, explain why the representing g satisfies 0≤g≤1 and why ν≪μ removes the set where g=1.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
If ν has density f relative to μ and ρ has density g relative to ν, prove ρ has density gf relative to μ.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
