# 7.9: Lebesgue–Stieltjes Measures

7.9: Lebesgue–Stieltjes Measures

## Lebesgue-stieltjes Measure on R ☆

It is well known that the notion of measure and integral were released early enough in close connection with practical problems of measuring of geometric figures. Notion of measure was outlined in the early 20th century through H. Lebesgue's research, founder of the modern theory of measure and integral. It was developed concurrently a technique of integration of functions. Gradually it was formed a specific area today called the measure and integral theory. Essential contributions to building this theory were made by a large number of mathematicians: C. Carathodory, J. Radon, O. Nikodym, S. Bochner, J. Pettis, P. Halmos and many others. In the following we present several abstract sets, classes of sets, positive measure on a cr-algebra, Lebesgue measure on R and the particularly measure Lebesgue-Stieltjes measure on R.

## Vector Integration in Banach Spaces and Application to Stochastic Integration

Nicolae Dinculeanu , in Handbook of Measure Theory , 2002

### 8.1. Processes with finite variation or semivariation

Let X : ℝ + × Ω → E ⊂ L ( F , G ) Let be a process.

DEFINITION. We say that the process X has finite variation (respectively finite semivariation relative to ( F, G)), if for every ω ∈ Ω. the path t ↦ Xt(ω) has finite variation (respectively finite semivariation relative to (F, G)) on each interval [0, t], or, equivalently, on each interval]− ∞, t].

We say that X has p-integrable variation |X| (respectively p-integrable semivariation X ¯ F . G ) if | X | ∞ ∈ L 1 (respectively ( X ¯ F . G ) ∞ ∈ L 1 ) .

Assume X is right continuous and has finite variation |X|. If X is measurable (respectively optional, predictable), then so is |X|.

Assume X is right continuous and has finite semivariation X ˜ F . G . Assume also that eitherF or G is separable. If X is measurable (respectively optional, predictable), then so is it X ˜ F . G

Let $C_1=left[0,frac<1><3> ight]cupleft[frac<2><3>,1 ight]$, $C_2=left[0,frac<1><9> ight]cupleft[frac<2><9>,frac<3><9> ight]cupleft[frac<6><9>,frac<7><9> ight]cupleft[frac<8><9>,frac<9><9> ight]$ and so on the usual sets used to define the Cantor set. Then $mu_F$ is the limit as $n o +infty$ of the probability measure $mu_$ on $C_n$. Let $I=[a,a+3b]$ be any closed interval of the real line and $J$ the same interval without its middle third, $J=[a,a+b]cup[a+2b,a+3b]$. Then: $int_I x^2 dmu = frac<1><3>left((a+3b)^3-a^3 ight)=3b(a^2+3ab+3b^2),$ $frac<3><2>int_J x^2 dmu = 3b(a^2+3ab+3b^2)+b^3,$ so: $frac<3><2>int_J x^2 dmu = int_I x^2 dmu + frac<27>, ag<1>$ giving immediately: $int_<0>^ <1>x^2, dmu_F = lim_int_<0>^ <1>x^2, dmu_ = lim_sum_^frac<1><3^<2k+1>>=color<8>> . ag<2>$

Here is another calculation, based on how $mu_F$ behaves under scaling $x o frac<1> <3>x$ and reflection $x o 1-x$ .

Let $M_n = int_0^1 x^n , dmu_F(x).$

Case $n=0$ is trivial: $M_0 = int_0^1 dmu_F(x) = 1.$

Note also that we have $int_0^ <1/3>dmu_F(x) = frac<1><2>.$

Case $n=1$ we solve by splitting the integral in two parts: $M_1 = int_0^1 x , dmu_F(x) = int_0^ <1/3>x , dmu_F(x) + int_<2/3>^1 x , dmu_F(x)$ The first term we rewrite using scaling: $int_0^ <1/3>x , dmu_F(x) = < x = frac<1> <3>y > = int_0^1 (frac<1><3>y) , dmu_F(frac<1><3>y) = int_0^1 frac<1><3>y , frac<1> <2>dmu_F(y) = frac<1> <6>int_0^1 y , dmu_F(y) = frac<1> <6>M_1$ and the second term using reflection: $int_<2/3>^1 x , dmu_F(x) = int_<1/3>^0 (1-z) , dmu_F(1-z) = int_0^ <1/3>(1-z) , dmu_F(z) = int_0^ <1/3>dmu_F(z) - int_0^ <1/3>z , dmu_F(z) = frac<1> <2>- frac<1> <6>M_1$ Thus, $M_1 = frac<1> <6>M_1 + left( frac<1> <2>- frac<1> <6>M_1 ight) = frac<1><2>.$

## Attempt to construct a uncountable measure zero set

A cantor set is a zero-set and uncountable and can be built as follows:consider interval $[0,1]in R$. Equipartition it by 3 subsets ,namely, $[0,1/3],[1/3,2/3],[2/3,1]$ and remove the middle sub-interval [1/3,2/3]. Perform the same procedure on remaining intervals $[0,1/3],[2/3,1]$ and construct intervals $[0,1/9],[2/9,1/3],[2/3,7/9],[8/9,1]$. Following this idea till infinity you will finally have an uncountable zero-set.

A classic example is Cantor's ternary set: $K_3=igglfrac<3^i>Bigm|forall ige 1, x_iin<0,2>iggr>$ i.e. the set of real numbers in $[0,1]$ which a base $3$ expansion without $1$ among the ternary digits.

Since your question seems to be about your choice of $A$ in particular, I'll address that instead of supplying a different option (which others have done very well).

The complement of your $A$ consists of all $x$ with only finitely many zeroes. All such $x$ are rational (because, for example, .10010111111ldots$is the same as .10011$). The set of all rationals has measure $, so the complement of$A$has measure zero. So$A$has (very) positive measure. Others have suggested the Cantor set instead, which would be an excellent option. ## What do different TSH levels mean? The thyroid stimulating hormone (TSH) test measures the amount of TSH in the blood. The results convey how well the thyroid is functioning. Doctors can use TSH test results to diagnose thyroid disorders, such as hypothyroidism and hyperthyroidism. The pituitary gland produces TSH, which is a hormone that stimulates the thyroid gland. The thyroid is a butterfly-shaped gland in the throat. It produces hormones that help regulate many bodily functions, such as metabolism, heart rate, and body temperature. In this article, we describe the TSH test and results. We also discuss what high and low TSH levels indicate and available treatments. Share on Pinterest A TSH test can help a doctor diagnose hypothyroidism and hyperthyroidism. The normal range depends on a person’s age and whether they are pregnant. The ranges tend to increase as a person gets older. Research has not shown a consistent difference in TSH levels between males and females. However, according to the American Thyroid Association, doctors generally consider levels to be within a normal range if they are between 0.4 and 4.0 milliunits per liter (mU/l). The table below provides estimates of TSH levels that are normal, low (indicating hyperthyroidism) and high (indicating hypothyroidism):  hyperthyroidism normal mild hypothyroidism hypothyroidism 0–0.4 0.4–4 4–10 10 Most labs use these reference values. However, there is some debate about these ranges — the author of a 2016 review has found that normal levels are more likely to fall between 0.5 and 2.5 milli-international units (mIU) per milliliter. Females are more likely to experience thyroid dysfunction than males. The Office on Women’s Health report that 1 in 8 females experience thyroid problems at some point. This includes hyperthyroidism and hypothyroidism. The risk of thyroid problems increases during pregnancy and around menopause. Research has not shown a consistent difference in TSH levels between males and females. A 2002 study reports higher TSH levels in females than in males, but a 2013 study reports that males have higher median TSH levels. It appears that any such difference is small, varies with age, and is unlikely to be clinically relevant. In some people, thyroid conditions are linked with sexual dysfunction. This may affect more males than females. According to a 2019 study, 59–63% of males who have hypothyroidism also experience sexual dysfunction, compared with 22–46% of females with hypothyroidism. ## Hospital-Acquired Condition (HAC) Reduction Program The HAC Reduction Program encourages hospitals to improve patients’ safety and reduce the number of conditions people experience from their time in a hospital, such as pressure sores and hip fractures after surgery. ### Why is the HAC Reduction Program important? The HAC Reduction Program encourages hospitals to improve patients’ safety and implement best practices to reduce their rates of infections associated with health care. ### Which hospitals do the HAC Reduction Program apply to? As defined under the Social Security Act, the HAC Reduction Program applies to all subsection (d) hospitals (that is, general acute care hospitals). Some hospitals and hospital units, such as the following, are exempt from the HAC Reduction Program: • Critical access hospitals • Rehabilitation hospitals and units • Long-term care hospitals • Psychiatric hospitals and units • Children’s hospitals • Prospective Payment System-exempt cancer hospitals • Veterans Affairs medical centers and hospitals • Short-term acute care hospitals located in U.S. territories (Guam, Puerto Rico, the U.S. Virgin Islands, the Northern Mariana Islands, and American Samoa) • Religious nonmedical health care institutions Note: For a full description of subsection (d) hospitals, refer to the Social Security Act on the Social Security Administration’s website at https://www.ssa.gov/OP_Home/ssact/ssact-toc.htm . Maryland hospitals are exempt from payment reductions under the HAC Reduction Program because they currently operate under a waiver agreement between CMS and the state of Maryland. ### What measures are included in the HAC Reduction Program? The following measures are included in the HAC Reduction Program, grouped here by category: Patient Safety and Adverse Events Composite (CMS PSI 90) We calculate the CMS PSI 90 using Medicare Fee-for-service claims. The CMS PSI 90 measure includes: • PSI 03 — Pressure Ulcer Rate • PSI 06 — Iatrogenic Pneumothorax Rate • PSI 08 — In Hospital Fall with Hip Fracture Rate • PSI 09 — Perioperative Hemorrhage or Hematoma Rate • PSI 10 — Postoperative Acute Kidney Injury Requiring Dialysis Rate • PSI 11 — Postoperative Respiratory Failure Rate • PSI 12 — Perioperative Pulmonary Embolism or Deep Vein Thrombosis Rate • PSI 13 — Postoperative Sepsis Rate • PSI 14 — Postoperative Wound Dehiscence Rate • PSI 15 — Unrecognized Abdominopelvic Accidental Puncture/Laceration Rate Centers for Disease Control and Prevention's National Healthcare Safety Network healthcare-associated infection (HAI) measures We calculate the following HAI measures using data on infections taken from charts, reports, and other sources and reported to the National Healthcare Safety Network : • Central Line-Associated Bloodstream Infection (CLABSI) • Catheter-Associated Urinary Tract Infection (CAUTI) • Surgical Site Infection (SSI) (for colon and abdominal hysterectomy procedures) • Methicillin-resistant Staphylococcus aureus (MRSA) bacteremia • Clostridium difficile Infection (CDI) ### How do payments change under the HAC Reduction Program? We reduce the payments of subsection (d) hospitals with a Total HAC Score greater than the 75th percentile of all Total HAC Scores (that is, the worst-performing quartile) by 1 percent . We first adjust payments for the Hospital Value-Based Purchasing Program, Hospital Readmissions Reduction Program, disproportionate share hospital payments, and indirect medical education payments based on the base-operating diagnosis-related group amount. Then, we apply the HAC Reduction Program payment reduction based on the overall Medicare payment amount . For example, if a hospital is subject to a 2-percent payment reduction for both the Hospital Readmissions Reduction Program and Hospital Value-Based Purchasing Programs, does not have disproportionate share hospital adjustments, does not have indirect medical education adjustments, and is subject to the HAC Reduction Program payment adjustment, then the final Medicare payment for a discharge with a$10,000 base-operating diagnosis-related group payment would be as follows:

Base-operating DRG amount: $10,000 Hospital Readmissions Reduction Program payment adjustment =$10,000 * -0.02 = -$200 Hospital Value-Based Purchasing Program payment adjustment =$10,000 * -0.02 = -$200 Disproportionate share hospital and indirect medical education payment adjustment = Overall Medicare payment amount =$10,000 - $200 -$200 = $9,600 HAC Reduction Program payment adjustment =$9,600 * -0.01 = -$96 Final Medicare payment =$9,600 - $96 =$9,504

More details on the Inpatient Prospective Payment System methodology are available in our Acute Payment System Fact Sheet (PDF).

### When do we adjust payments under the HAC Reduction Program?

We adjust payments when we pay hospital claims. The payment reduction is for all Medicare fee-for-service discharges in the corresponding fiscal year. We let hospitals know whether their payment will be reduced in a HAC Reduction Program Hospital-Specific Report, which is delivered to hospitals through the QualityNet Secure Portal.

More information is available in the QualityNet HAC Reduction Program Scoring Methodology section .

### What is the Scoring Calculations Review and Correction period for the HAC Reduction Program?

The FY 2014 Inpatient Prospective Payment System/Long-Term Care Hospital Prospective Payment System (IPPS/LTCH PPS) Final Rule requires CMS to give hospitals confidential Hospital-Specific Reports. We give hospitals 30 days to review their HAC Reduction Program data, submit questions about the calculation of their results, and request corrections before public reporting .

The Scoring Calculations Review and Corrections period let hospitals request corrections to the following:

• CMS PSI 90 measure result
• Measure score for each measure in the program
• Total HAC Score
• Payment reduction status

The Scoring Calculation Review and Corrections period does not let hospitals:

• Submit more corrections to the underlying CMS PSI 90 claims data
• Add new claims to the data extract we use to calculate the results
• Correct reported number of healthcare–associated infections
• Correct standardized infection ratios
• Correct reported central-line days, urinary catheter days, surgical procedures performed, or patient days

### How will I know if the HAC Reduction Program changes?

Changes to the program happen through rulemaking and are published every year after a public comment period. They’ll be proposed in the IPPS/LTCH PPS Proposed Rule and finalized in the IPPS/LTCH PPS Final Rule .

Detailed program information can be found on the HAC Reduction Program pages of QualityNet .

## Millimeter to Inch Conversion Table

Millimeter measurements converted to inches in decimal and fraction form
Millimeters Inches (decimal) Inches (fraction)
1 mm 0.03937" 3/64"
2 mm 0.07874" 5/64"
3 mm 0.11811" 1/8"
4 mm 0.15748" 5/32"
5 mm 0.19685" 13/64"
6 mm 0.23622" 15/64"
7 mm 0.275591" 9/32"
8 mm 0.314961" 5/16"
9 mm 0.354331" 23/64"
10 mm 0.393701" 25/64"
11 mm 0.433071" 7/16"
12 mm 0.472441" 15/32"
13 mm 0.511811" 33/64"
14 mm 0.551181" 35/64"
15 mm 0.590551" 19/32"
16 mm 0.629921" 5/8"
17 mm 0.669291" 43/64"
18 mm 0.708661" 45/64"
19 mm 0.748031" 3/4"
20 mm 0.787402" 25/32"
21 mm 0.826772" 53/64"
22 mm 0.866142" 55/64"
23 mm 0.905512" 29/32"
24 mm 0.944882" 15/16"
25 mm 0.984252" 63/64"
26 mm 1.0236" 1 1/32"
27 mm 1.063" 1 1/16"
28 mm 1.1024" 1 7/64"
29 mm 1.1417" 1 9/64"
30 mm 1.1811" 1 3/16"
31 mm 1.2205" 1 7/32"
32 mm 1.2598" 1 17/64"
33 mm 1.2992" 1 19/64"
34 mm 1.3386" 1 11/32"
35 mm 1.378" 1 3/8"
36 mm 1.4173" 1 27/64"
37 mm 1.4567" 1 29/64"
38 mm 1.4961" 1 1/2"
39 mm 1.5354" 1 17/32"
40 mm 1.5748" 1 37/64"

## 7.9: Lebesgue–Stieltjes Measures

Earthquake Magnitude Scale

 Magnitude Earthquake Effects Estimated NumberEach Year 2.5 or less Usually not felt, but can be recorded by seismograph. 900,000 2.5 to 5.4 Often felt, but only causes minor damage. 30,000 5.5 to 6.0 Slight damage to buildings and other structures. 500 6.1 to 6.9 May cause a lot of damage in very populated areas. 100 7.0 to 7.9 Major earthquake. Serious damage. 20 8.0 or greater Great earthquake. Can totally destroy communities near the epicenter. One every 5 to 10 years

Earthquake Magnitude Classes

Earthquakes are also classified in categories ranging from minor to great, depending on their magnitude.

Preface
Chapter 1. Orientation
1.1 Real Numbers
1.2 Sets
1.3 Mathematical Induction
1.4 Logic
1.5 Completion
1.6 Dedekind Cuts
1.7 Geometry
1.8 Decimal, Ternary, and Binary Representations
1.9 Absolute Value and Inequalities
1.10 Complex Numbers
Chapter 2. Sets and Spaces
2.1 Equivalent Sets
2.2 Infinite Sets
2.3 Sequences of Sets
2.4 Metric Spaces
2.5 Open Sets
2.6 Compact Sets
2.7 Properties in Ek
2.8 Perfect Sets in Ek
2.9 Cardinals
2.10 Connected Sets
2.11 Distance and Relative Properties
2.12 Ek as a Vector Space
Chapter 3. Sequences and Series
3.1 The Extended Real Number System
3.2 Limits Inferior and Superior
3.3 Limit of a Real Sequence
3.4 Sequences in Metric Spaces
3.5 Sequences of Complex Numbers
3.6 Series
3.7 Series of Real Numbers
3.8 Rearranging and Grouping
3.9 Absolute and Conditional Convergence
3.10 The Space ℓ2
3.11 Double Sequences and Series
3.12 Power Series
Chapter 4. Measure and Integration
4.1 Outer Lebesgue Measure in E1
4.2 Outer Measures and Measurability
4.3 Measurable Functions
4.5 Lebesgue Integration
4.6 Real ℓ2 Spaces
4.7 Complex ℓ2 Spaces
Chapter 5. Measure Theory
5.1 Metric Outer Measure
5.2 Properties of Lebesgue Measure
5.3 σ-Algebras
5.4 Lebesgue Outer k-Measure
5.5 Fubini's Theorem
5.6 Outer Ordinate Sets
5.7 Ergodic Theory
Chapter 6. Continuity
6.1 Limits and Continuity of Functions
6.2 Relative Openness and Continuity
6.3 Uniformity
6.4 Weierstrass Approximation Theorem
6.5 Absolute Continuity
6.6 Equicontinuity
6.7 Semicontinuity
6.8 Discontinuities
6.9 Approximate Continuity
6.10 Continuous Linear Functionals
Chapter 7. Derivatives
7.1 Dini Derivatives
7.2 Mean Values
7.3 Trigonometry
7.4 Fourier Series
7.5 Derivatives Almost Everywhere
7.6 Bounded Variation
7.7 Derivatives and Integrals
7.8 Change of Variable
7.9 Exponents and Logarithms
7.10 Taylor's Theorem
Chapter 8. Stieltjes Integrals
8.1 Riemann-Stieltjes Integrals
8.2 Darboux-Stieltjes Integrals
8.3 Riemann Integrals
8.4 Integrators of Bounded Variation
8.5 Lebesgue Integral Relations
8.6 Lebesgue-Stieltjes Integrals
8.7 Lebesgue Decomposition and Radon-Nikodym Theorems
Bibliography
Index