Measure and integration: construction, convergence and probability
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Suggested session length: 300 minutes, as an unvalidated planning estimate. This collection contains 12 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Attempt these public mixed problems before revealing their guidance. State the measure spaces, measurable representatives, finiteness assumptions and every theorem used.
Calculations and saved responses do not certify proof quality or independent examination performance. Written work requires human mathematical review.
The work is untimed; the suggested duration supports planning. Revise a failed argument after returning to its mapped lessons, and distinguish proved steps from explicitly named dependencies.
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For n≥1 let μ_n put mass1−1/n at0 and mass1/n uniformly on[−n²,n²]. Compute its characteristic function, weak limit and second moment. Prove the whole family is tight despite the escaping component. Contrast it with the pure uniform law on[−n²,n²], and explain why the continuity theorem does not imply convergence of second moments. Identify the proof mechanisms that turn the transform limit into a unique probability limit.
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.
Let X have P(X>x)=(2/x)^(4/3) for x≥2, with support[2,∞). Derive its mean and second-moment behavior. At cutoff16 compute discard probability, zero-truncated mean, clipped mean and truncated second moment. State and justify which strong law applies to iid copies, including the core weighted-variance and centered-series argument. Then set every observation equal to the same X and diagnose the failure. Does an almost-sure law supply a universal deterministic finite sample size after which every path stays near the mean?
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.
On Ω={1,2,3,4,5,6,7,8}, observe only membership in A={1,2,3,4} and B={2,4,6,8}. Find the atoms and size of σ(A,B). Give the four atoms masses 1,2,3,4 in the order A∩B, A outside B, B outside A, and outside both. Define an outer measure on arbitrary subsets as the infimum of costs of countable covers by measurable sets. Compute the outer measure of {2,5}, and determine whether {2} is Carathéodory measurable. Then prove continuity from above for a general measure when the first set has finite measure, and give a counterexample without that hypothesis.
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.
On [0,2], assign a half-open interval (a,b] the value b²−a², and assign {0} value zero. Explain how this defines a finite premeasure on the algebra of finite disjoint interval unions and the endpoint atom, and why its extension to the Borel sets is unique. Identify the extension using a length density. Compute the measure of (1/2,3/2] and of the rational points. Explain the completed measure and its null sets. Finally, for f(x)=x, compute the uncapped dyadic lower approximation at x=7/5 with mesh 1/8, and prove that capped dyadic approximations increase to any nonnegative measurable f.
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.
Use length measure on (0,1). For n≥1 let u_n=n·1_(0,n^(-3)) and v_n=n³·1_(0,n^(-3)). Compute their integrals and pointwise limits. Find one integrable majorant for the entire sequence u_n, and prove that no such majorant exists for v_n. Explain Fatou’s inequality for each sequence. Separately, let E_k=(2^(-k),2^(-(k-1))] and h=Σ_(k≥1) k·1_(E_k). Derive the integral of h from the definition for simple functions and monotone convergence, including the value of the numerical series.
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.
On a probability space, measurable X_n converge in probability to measurable X and sup_n E|X_n|^(3/2)≤8. Prove X is integrable and X_n→X in L1 without assuming almost-sure convergence of the whole sequence. Quantify the common large-value tail bound. Extract an almost-surely convergent subsequence and explain how Egoroff improves that subsequence off a set of probability below ε. If the space is [0,1] with Lebesgue probability, explain how to obtain a compact set of measure above 1−ε on which X is continuous and the selected subsequence converges uniformly. State why the compact-set claim is not an assertion that every X_n is continuous there.
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.
On the triangle 0<y<x<3, a joint law has density cx relative to planar length, zero elsewhere. Find c, both marginal densities, E(X), and E(Y). Let U=Y/X on the positive-support triangle. Prove that U is uniform on (0,1) and independent of X by calculating joint rectangle probabilities, rather than merely asserting a Jacobian. Identify precisely where Tonelli, Fubini and uniqueness of product measures justify your steps. Explain how changing the density on the line x=0 affects this law.
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.
On (0,1) set f(x)=x^(-1/4) and g(x)=x^(-1/2). Determine membership of each in L2 and L3 where applicable, compute ∫fg, and verify Hölder with conjugate exponents 3 and 3/2. Next, on an arbitrary measure space assume h_k∈L3 and ||h_k||_3≤3^(-k) for k≥1. Prove the series converges absolutely almost everywhere and in L3, with an explicit norm bound for the tail from k=m. Finally explain how to approximate the resulting L3 class by finite-valued simple functions of finite-measure support, without assuming the ambient measure is finite.
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.
On [0,1], let ν(E)=∫_E(4x−1)dx+2δ_0(E)−3δ_1(E), where δ_a is unit point mass at a. Find a Hahn decomposition, ν⁺([0,1]), ν⁻([0,1]) and total variation. Decompose ν into absolutely continuous and singular signed parts relative to length. Then find dν/dμ for μ=length+δ_0+δ_1, explaining why the density’s endpoint values differ from 4x−1. Prove uniqueness almost everywhere of the density using measurable comparison sets, and explain why a singular measure need not be atomic.
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.
Let f(x)=1/(2sqrt(x)) on (0,1), and F(x)=∫_0^x f(t)dt on [0,1]. Compute F and prove absolute continuity despite the unbounded derivative near zero. Show F is not Lipschitz. Derive the one-dimensional weak maximal estimate with factor three using a finite interval selection argument, and explain how this estimate plus continuous L1 approximation proves differentiation almost everywhere. Finally contrast F with the Cantor distribution function to explain why continuity and an almost-everywhere derivative do not suffice for the integral formula.
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.
Four outcomes have probabilities (1,2,3,4)/10 and X values (−2,1,3,7). Information G distinguishes only the first two outcomes from the last two. Compute E(X given G), E(X), the residual mean square and the variance of the conditional mean, and verify total variance. Construct conditional expectation abstractly by Radon–Nikodym on G and prove its L2 optimality. Finally adjoin an independent fair bit B. Prove that observing B in addition to G does not change the optimal predictor in this product model; explain why this conclusion would fail if B carried information about the outcome.
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.
Let X_i be iid with P(X_i=−1)=2/3 and P(X_i=2)=1/3. Compute the mean and variance. Give a Chebyshev bound for a sample-mean error of at least 1/4 at n=512. Prove almost-sure convergence of the sample means using first-crossing maximal estimates and dyadic times, rather than summing the divergent 1/n bound. Derive the characteristic-function limit of S_n/sqrt(2n), where S_n=ΣX_i, and name the theorem required to infer weak convergence. Explain why neither the strong law nor that weak limit says every finite standardized sum is normally distributed.
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.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
For n≥1 let μ_n put mass1−1/n at0 and mass1/n uniformly on[−n²,n²]. Compute its characteristic function, weak limit and second moment. Prove the whole family is tight despite the escaping component. Contrast it with the pure uniform law on[−n²,n²], and explain why the continuity theorem does not imply convergence of second moments. Identify the proof mechanisms that turn the transform limit into a unique probability limit.
Let X have P(X>x)=(2/x)^(4/3) for x≥2, with support[2,∞). Derive its mean and second-moment behavior. At cutoff16 compute discard probability, zero-truncated mean, clipped mean and truncated second moment. State and justify which strong law applies to iid copies, including the core weighted-variance and centered-series argument. Then set every observation equal to the same X and diagnose the failure. Does an almost-sure law supply a universal deterministic finite sample size after which every path stays near the mean?
On Ω={1,2,3,4,5,6,7,8}, observe only membership in A={1,2,3,4} and B={2,4,6,8}. Find the atoms and size of σ(A,B). Give the four atoms masses 1,2,3,4 in the order A∩B, A outside B, B outside A, and outside both. Define an outer measure on arbitrary subsets as the infimum of costs of countable covers by measurable sets. Compute the outer measure of {2,5}, and determine whether {2} is Carathéodory measurable. Then prove continuity from above for a general measure when the first set has finite measure, and give a counterexample without that hypothesis.
On [0,2], assign a half-open interval (a,b] the value b²−a², and assign {0} value zero. Explain how this defines a finite premeasure on the algebra of finite disjoint interval unions and the endpoint atom, and why its extension to the Borel sets is unique. Identify the extension using a length density. Compute the measure of (1/2,3/2] and of the rational points. Explain the completed measure and its null sets. Finally, for f(x)=x, compute the uncapped dyadic lower approximation at x=7/5 with mesh 1/8, and prove that capped dyadic approximations increase to any nonnegative measurable f.
Use length measure on (0,1). For n≥1 let u_n=n·1_(0,n^(-3)) and v_n=n³·1_(0,n^(-3)). Compute their integrals and pointwise limits. Find one integrable majorant for the entire sequence u_n, and prove that no such majorant exists for v_n. Explain Fatou’s inequality for each sequence. Separately, let E_k=(2^(-k),2^(-(k-1))] and h=Σ_(k≥1) k·1_(E_k). Derive the integral of h from the definition for simple functions and monotone convergence, including the value of the numerical series.
On a probability space, measurable X_n converge in probability to measurable X and sup_n E|X_n|^(3/2)≤8. Prove X is integrable and X_n→X in L1 without assuming almost-sure convergence of the whole sequence. Quantify the common large-value tail bound. Extract an almost-surely convergent subsequence and explain how Egoroff improves that subsequence off a set of probability below ε. If the space is [0,1] with Lebesgue probability, explain how to obtain a compact set of measure above 1−ε on which X is continuous and the selected subsequence converges uniformly. State why the compact-set claim is not an assertion that every X_n is continuous there.
On the triangle 0<y<x<3, a joint law has density cx relative to planar length, zero elsewhere. Find c, both marginal densities, E(X), and E(Y). Let U=Y/X on the positive-support triangle. Prove that U is uniform on (0,1) and independent of X by calculating joint rectangle probabilities, rather than merely asserting a Jacobian. Identify precisely where Tonelli, Fubini and uniqueness of product measures justify your steps. Explain how changing the density on the line x=0 affects this law.
On (0,1) set f(x)=x^(-1/4) and g(x)=x^(-1/2). Determine membership of each in L2 and L3 where applicable, compute ∫fg, and verify Hölder with conjugate exponents 3 and 3/2. Next, on an arbitrary measure space assume h_k∈L3 and ||h_k||_3≤3^(-k) for k≥1. Prove the series converges absolutely almost everywhere and in L3, with an explicit norm bound for the tail from k=m. Finally explain how to approximate the resulting L3 class by finite-valued simple functions of finite-measure support, without assuming the ambient measure is finite.
On [0,1], let ν(E)=∫_E(4x−1)dx+2δ_0(E)−3δ_1(E), where δ_a is unit point mass at a. Find a Hahn decomposition, ν⁺([0,1]), ν⁻([0,1]) and total variation. Decompose ν into absolutely continuous and singular signed parts relative to length. Then find dν/dμ for μ=length+δ_0+δ_1, explaining why the density’s endpoint values differ from 4x−1. Prove uniqueness almost everywhere of the density using measurable comparison sets, and explain why a singular measure need not be atomic.
Let f(x)=1/(2sqrt(x)) on (0,1), and F(x)=∫_0^x f(t)dt on [0,1]. Compute F and prove absolute continuity despite the unbounded derivative near zero. Show F is not Lipschitz. Derive the one-dimensional weak maximal estimate with factor three using a finite interval selection argument, and explain how this estimate plus continuous L1 approximation proves differentiation almost everywhere. Finally contrast F with the Cantor distribution function to explain why continuity and an almost-everywhere derivative do not suffice for the integral formula.
Four outcomes have probabilities (1,2,3,4)/10 and X values (−2,1,3,7). Information G distinguishes only the first two outcomes from the last two. Compute E(X given G), E(X), the residual mean square and the variance of the conditional mean, and verify total variance. Construct conditional expectation abstractly by Radon–Nikodym on G and prove its L2 optimality. Finally adjoin an independent fair bit B. Prove that observing B in addition to G does not change the optimal predictor in this product model; explain why this conclusion would fail if B carried information about the outcome.
Let X_i be iid with P(X_i=−1)=2/3 and P(X_i=2)=1/3. Compute the mean and variance. Give a Chebyshev bound for a sample-mean error of at least 1/4 at n=512. Prove almost-sure convergence of the sample means using first-crossing maximal estimates and dyadic times, rather than summing the divergent 1/n bound. Derive the characteristic-function limit of S_n/sqrt(2n), where S_n=ΣX_i, and name the theorem required to infer weak convergence. Explain why neither the strong law nor that weak limit says every finite standardized sum is normally distributed.
