# Chapter 2, Measure Theory Paul R. Halmos

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#### Result

If E is a set contains each Ei for 1≤ i ≤n, then the measure of E is greater or equal to the sum of all the measures of Ei. Note that this holds only for both E and Ei’s are bounded, left closed, and right open intervals. It may fail to be true in the more general case.

#### Reference

Measure Theory, Paul R. Halmos, Chapter 2 Section 8

#### Related Topics

• Definitions
• Theorems/Corollary/Propositions/Lemmas

### Theorem 8.2

#### Result

The measure of a closed interval is strictly smaller than the sum of the measures of its finite cover generated by bounded open intervals.

#### Reference

Measure Theory, Paul R. Halmos, Chapter 2 Section 8

### Theorem 8.3

#### Result

The measure is monotone.

#### Reference

Measure Theory, Paul R. Halmos, Chapter 2 Section 8

#### Related Topics

• Definitions
• Theorems/Corollary/Propositions/Lemmas

### Theorem 8.4

#### Result

measure is a countably additive set function

#### Reference

Measure Theory, Paul R. Halmos, Chapter 2 Section 8

### Theorem 8.5

#### Result

Existence of finite measure

#### Reference

Measure Theory, Paul R. Halmos, Chapter 2 Section 8

#### Related Topics

• Definitions
• Theorems/Corollary/Propositions/Lemmas

### Theorem 9.1

#### Result

A measure on a ring is monotone and subtractive

#### Reference

Measure Theory, Paul R. Halmos, Chapter 2 Section 9

#### Related Topics

• Definitions
• Monotoneness
• Theorems/Corollary/Propositions/Lemmas

### Theorem 9.2

#### Result

Measure on a ring preserves ≤.

#### Reference

Measure Theory, Paul R. Halmos, Chapter 2 Section 9

#### Related Topics

• Definitions
• Theorems/Corollary/Propositions/Lemmas

### Theorem 9.3

#### Result

Measure on a ring preserves ≥.

#### Reference

Measure Theory, Paul R. Halmos, Chapter 2 Section 9

#### Related Topics

• Definitions
• Theorems/Corollary/Propositions/Lemmas

### Theorem 9.4

#### Result

Measure on a ring preserves the limit of increasing sequence of sets in such ring.

#### Reference

Measure Theory, Paul R. Halmos, Chapter 2 Section 9

#### Related Topics

• Definitions
• Theorems/Corollary/Propositions/Lemmas

### Theorem 9.5

#### Result

Measure on a ring preserves the limit of decreasing sequence of sets in such ring

#### Reference

Measure Theory, Paul R. Halmos, Chapter 2 Section 9

#### Related Topics

• Definitions
• Theorems/Corollary/Propositions/Lemmas

### Theorem 9.6

#### Result

Construction from set function to measurer on a ring. If the set function on a ring is finite, nonnegative, additive, and continuous from above at 0 or continuous from below in this set, then it is sufficient to build a measure from such a set function.

#### Reference

Measure Theory, Paul R. Halmos, Chapter 2 Section 9

#### Related Topics

• Theorems/Corollary/Propositions/Lemmas

### Theorem 10.1

#### Result

This theorem shows us how to build an outer measure on the hereditary sigma-ring from a measure on a ring. Indeed, the outer measure on a hereditary sigma-ring is an extension of the measure on the ring. Further, the total sigma-finite property is preserved in such an extension.

#### Reference

Measure Theory, Paul R. Halmos, Chapter 2 Section 10

#### Related Topics

• Definitions
• Theorems/Corollary/Propositions/Lemmas

### Theorem 11.1

#### Result

This theorem provides a way to construct a ring from mu*-measurable sets on a hereditary sigma-ring. In the next Theorem, we make a further assumption that this construction of ring is in fact a Sigma-Ring.

#### Reference

Measure Theory, Paul R. Halmos, Chapter 2 Section 8

#### Related Topics

• Definitions
• Theorems/Corollary/Propositions/Lemmas

### Theorem 11.2

#### Result

The first part of the theorem gives a further assumption to the previous theorem. While the second part declares that: if a set in sigma-ring could be expressed as a countable union of disjoint subsets in the sigma-ring, then the outer measure of the intersection between an arbitrary set in the sigma-ring H and such a set is in fact the countable sum of outer measure throughout each intersection between that arbitrary set and the disjoint sets.

#### Reference

Measure Theory, Paul R. Halmos, Chapter 2 Section 8

#### Related Topics

• Definitions
• Monotoneness
• Theorems/Corollary/Propositions/Lemmas

### Theorem 11.3

#### Result

Methodology in building up a complete measure in a mu* measurable set from a hereditary sigma-ring by along with an outer measure.

#### Reference

Measure Theory, Paul R. Halmos, Chapter 2 Section 8

#### Related Topics

• Definitions
• Theorems/Corollary/Propositions/Lemmas