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GRA:010117On the finite-cofinite algebra of N, geometric atoms 2^{-n} extend uniquely with mu(odd)=2/3, mu(3N)=1/7, mu(n>=5)=1/16

Graduate Real Analysis 1 Finite Cofinite PremeasureGraduate Real Analysis 1 Geometric Atom Extension

Sigma-algebras, measures, and outer-measure constructionGraduate Real Analysis I · Mixed review

GRA:010122Counting tails E_n={n,n+1,...} stay infinite while rho(E_n)=2^{1-n} and rho(E_4)=1/8

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Graduate Real Analysis 1 Continuity From AboveGraduate Real Analysis 1 Finite First Set Hypothesis

Sigma-algebras, measures, and outer-measure constructionGraduate Real Analysis I · Mixed review

GRA:010218Dyadic floor s_n=2^{-n} floor(2^n x) on [0,1] has integral 1/2-2^{-n-1} and at n=3 the ledger 7/16, 1/16, 1/8

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Graduate Real Analysis 1 Dyadic Floor MeasurabilityGraduate Real Analysis 1 Simple Function L1 Linf Gap

Measurable functions and modes of convergenceGraduate Real Analysis I · Mixed review

GRA:010223Rational sign f_n=nx/(1+n|x|) has L1 gap 2ln(1+n)/n, L2^2=2/(n+1), and n=3 values 4ln2/3 and 1/2

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Graduate Real Analysis 1 Pointwise Rational Sign LimitGraduate Real Analysis 1 LP Gap Norms Level Sets

Measurable functions and modes of convergenceGraduate Real Analysis I · Mixed review

GRA:010229Pushforward of Uniform[0,1] by x|->x^2 has CDF sqrt(t), density 1/(2sqrt t), mass 1/3 on [1/9,4/9], mean 1/3, variance 4/45

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Graduate Real Analysis 1 Square Map PushforwardGraduate Real Analysis 1 Pushforward Moments Variance

Measurable functions and modes of convergenceGraduate Real Analysis I · Mixed review

GRA:010319Layer-cake of x^{-1/5} on (0,1) yields int f=5/4, int f^2=5/3, weak-L5 quasi-norm 1, and a logarithmic L5 failure

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Graduate Real Analysis 1 Layer Cake RepresentationGraduate Real Analysis 1 Weak LP Versus LP

Lebesgue integration and convergence theoremsGraduate Real Analysis I · Mixed review

GRA:010324Logarithmic moments I_{a,m}=m!/(a+1)^{m+1} give I_{2,3}=2/27 and I_{-1/2,2}=16

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Graduate Real Analysis 1 Logarithmic Moment EvaluationGraduate Real Analysis 1 Gamma Substitution Recurrence

Lebesgue integration and convergence theoremsGraduate Real Analysis I · Mixed review

GRA:010330Monotone truncations min(n,x^{-1/2}) on (0,1] have integral 2-1/n, L1 tail 1/n, square integral 1+2ln n, and L2-divergent limit

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Graduate Real Analysis 1 Monotone Truncation IntegralsGraduate Real Analysis 1 Log Square L2 Divergence

Lebesgue integration and convergence theoremsGraduate Real Analysis I · Mixed review

GRA:010420The two-step ledger f=2 on [0,1/4] and f=-1 on (1/4,1] has integral -1/4, L1=5/4, L2=sqrt(7)/2, Linf=2

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Graduate Real Analysis 1 Step Function LP NormsGraduate Real Analysis 1 Distribution Function Ledger

Lp spaces, inequalities, and completenessGraduate Real Analysis I · Mixed review

GRA:010425x^{-2/5} is L^p on (0,1) iff p<5/2 with L2=sqrt(5), and on (1,infty) iff p>5/2 with L3=5^{1/3}

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Graduate Real Analysis 1 LP Near Zero IntegrabilityGraduate Real Analysis 1 LP AT Infinity Reversal

Lp spaces, inequalities, and completenessGraduate Real Analysis I · Mixed review

GRA:010521Alternating atoms (-1)^n 3^{-n} have total mass -1/4, variation 1/2, and nu({1..4})=-20/81

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Graduate Real Analysis 1 Atomic Signed MeasureGraduate Real Analysis 1 Total Variation Geometric

Signed measures and Radon-Nikodym theoryGraduate Real Analysis I · Mixed review

GRA:010527Oscillatory nu_n(E)=int_E sin(nx) on [0,2pi] has total mass 0, variation 4, window mass 1/n, and C1 bound O(1/n)

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Graduate Real Analysis 1 Oscillatory Signed VariationGraduate Real Analysis 1 C1 Integration BY Parts Bound

Signed measures and Radon-Nikodym theoryGraduate Real Analysis I · Mixed review

GRA:010626|x-y|^{-alpha} on the unit square is finite iff 0<alpha<1, equals 2/[(1-alpha)(2-alpha)], and is 8/3 at alpha=1/2

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Graduate Real Analysis 1 Diagonal Kernel TonelliGraduate Real Analysis 1 Power Integrability Threshold

Product measures, Tonelli, and FubiniGraduate Real Analysis I · Mixed review

GRA:010628The tilted triangle x+2y<=1 in the first quadrant has area 1/4 and section moments int x=1/12, int y=1/24, int(x+y)=1/8

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Graduate Real Analysis 1 Tilted Triangle FubiniGraduate Real Analysis 1 Section Moment Ledger

Product measures, Tonelli, and FubiniGraduate Real Analysis I · Mixed review

GRA:010631The hyperbolic sublevel xy<=a in the unit square has area a(1-ln a); at a=1/4 the area is (1+2ln 2)/4 and the complement is (3-2ln 2)/4

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Graduate Real Analysis 1 Hyperbolic Sublevel FubiniGraduate Real Analysis 1 Unit Square Log Area

Product measures, Tonelli, and FubiniGraduate Real Analysis I · Mixed review

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GRA:010117On the finite-cofinite algebra of N, geometric atoms 2^{-n} extend uniquely with mu(odd)=2/3, mu(3N)=1/7, mu(n>=5)=1/16

GRA:010122Counting tails E_n={n,n+1,...} stay infinite while rho(E_n)=2^{1-n} and rho(E_4)=1/8

GRA:010218Dyadic floor s_n=2^{-n} floor(2^n x) on [0,1] has integral 1/2-2^{-n-1} and at n=3 the ledger 7/16, 1/16, 1/8

GRA:010223Rational sign f_n=nx/(1+n|x|) has L1 gap 2ln(1+n)/n, L2^2=2/(n+1), and n=3 values 4ln2/3 and 1/2

GRA:010229Pushforward of Uniform[0,1] by x|->x^2 has CDF sqrt(t), density 1/(2sqrt t), mass 1/3 on [1/9,4/9], mean 1/3, variance 4/45

GRA:010319Layer-cake of x^{-1/5} on (0,1) yields int f=5/4, int f^2=5/3, weak-L5 quasi-norm 1, and a logarithmic L5 failure

GRA:010324Logarithmic moments I_{a,m}=m!/(a+1)^{m+1} give I_{2,3}=2/27 and I_{-1/2,2}=16

GRA:010330Monotone truncations min(n,x^{-1/2}) on (0,1] have integral 2-1/n, L1 tail 1/n, square integral 1+2ln n, and L2-divergent limit

GRA:010420The two-step ledger f=2 on [0,1/4] and f=-1 on (1/4,1] has integral -1/4, L1=5/4, L2=sqrt(7)/2, Linf=2

GRA:010425x^{-2/5} is L^p on (0,1) iff p<5/2 with L2=sqrt(5), and on (1,infty) iff p>5/2 with L3=5^{1/3}

GRA:010521Alternating atoms (-1)^n 3^{-n} have total mass -1/4, variation 1/2, and nu({1..4})=-20/81

GRA:010527Oscillatory nu_n(E)=int_E sin(nx) on [0,2pi] has total mass 0, variation 4, window mass 1/n, and C1 bound O(1/n)

GRA:010626|x-y|^{-alpha} on the unit square is finite iff 0<alpha<1, equals 2/[(1-alpha)(2-alpha)], and is 8/3 at alpha=1/2

GRA:010628The tilted triangle x+2y<=1 in the first quadrant has area 1/4 and section moments int x=1/12, int y=1/24, int(x+y)=1/8

GRA:010631The hyperbolic sublevel xy<=a in the unit square has area a(1-ln a); at a=1/4 the area is (1+2ln 2)/4 and the complement is (3-2ln 2)/4

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Problem results

GRA:010117On the finite-cofinite algebra of N, geometric atoms 2^{-n} extend uniquely with mu(odd)=2/3, mu(3N)=1/7, mu(n>=5)=1/16

GRA:010122Counting tails E_n={n,n+1,...} stay infinite while rho(E_n)=2^{1-n} and rho(E_4)=1/8

GRA:010218Dyadic floor s_n=2^{-n} floor(2^n x) on [0,1] has integral 1/2-2^{-n-1} and at n=3 the ledger 7/16, 1/16, 1/8

GRA:010223Rational sign f_n=nx/(1+n|x|) has L1 gap 2ln(1+n)/n, L2^2=2/(n+1), and n=3 values 4ln2/3 and 1/2

GRA:010229Pushforward of Uniform[0,1] by x|->x^2 has CDF sqrt(t), density 1/(2sqrt t), mass 1/3 on [1/9,4/9], mean 1/3, variance 4/45

GRA:010319Layer-cake of x^{-1/5} on (0,1) yields int f=5/4, int f^2=5/3, weak-L5 quasi-norm 1, and a logarithmic L5 failure

GRA:010324Logarithmic moments I_{a,m}=m!/(a+1)^{m+1} give I_{2,3}=2/27 and I_{-1/2,2}=16

GRA:010330Monotone truncations min(n,x^{-1/2}) on (0,1] have integral 2-1/n, L1 tail 1/n, square integral 1+2ln n, and L2-divergent limit

GRA:010420The two-step ledger f=2 on [0,1/4] and f=-1 on (1/4,1] has integral -1/4, L1=5/4, L2=sqrt(7)/2, Linf=2

GRA:010425x^{-2/5} is L^p on (0,1) iff p<5/2 with L2=sqrt(5), and on (1,infty) iff p>5/2 with L3=5^{1/3}

GRA:010521Alternating atoms (-1)^n 3^{-n} have total mass -1/4, variation 1/2, and nu({1..4})=-20/81

GRA:010527Oscillatory nu_n(E)=int_E sin(nx) on [0,2pi] has total mass 0, variation 4, window mass 1/n, and C1 bound O(1/n)

GRA:010626|x-y|^{-alpha} on the unit square is finite iff 0<alpha<1, equals 2/[(1-alpha)(2-alpha)], and is 8/3 at alpha=1/2

GRA:010628The tilted triangle x+2y<=1 in the first quadrant has area 1/4 and section moments int x=1/12, int y=1/24, int(x+y)=1/8

GRA:010631The hyperbolic sublevel xy<=a in the unit square has area a(1-ln a); at a=1/4 the area is (1+2ln 2)/4 and the complement is (3-2ln 2)/4

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GRA:010117On the finite-cofinite algebra of N, geometric atoms 2^{-n} extend uniquely with mu(odd)=2/3, mu(3N)=1/7, mu(n>=5)=1/16View problem statement

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GRA:010117On the finite-cofinite algebra of N, geometric atoms 2^{-n} extend uniquely with mu(odd)=2/3, mu(3N)=1/7, mu(n>=5)=1/16View problem statement

GRA:010117On the finite-cofinite algebra of N, geometric atoms 2^{-n} extend uniquely with mu(odd)=2/3, mu(3N)=1/7, mu(n>=5)=1/16

GRA:010117On the finite-cofinite algebra of N, geometric atoms 2^{-n} extend uniquely with mu(odd)=2/3, mu(3N)=1/7, mu(n>=5)=1/16