Last week 1. We showed that is -additive on M . 2. Hence | M is - - PowerPoint PPT Presentation

Last week

• 1. We showed that λ∗ is σ-additive on M.
• 2. Hence λ∗|M is a measure! We denote it simply by λ.
• 3. We started on general measure and (a little) integration theory.
• 4. You proved that all measures have nice properties.
• 5. You looked at the concepts of almost everywhere and complete

measures.

• 6. We defined the notion of an A-measurable function.
• 7. We defined a simple A-measurable function.
• 8. We defined the integral of a non-negative simple A-measurable

function.

• 9. We didn’t quite show that the definition was independent of our

representation of the simple function.

Last week

• 1. We showed that λ∗ is σ-additive on M.
• 2. Hence λ∗|M is a measure! We denote it simply by λ.
• 3. We started on general measure and (a little) integration theory.
• 4. You proved that all measures have nice properties.
• 5. You looked at the concepts of almost everywhere and complete

measures.

• 6. We defined the notion of an A-measurable function.
• 7. We defined a simple A-measurable function.
• 8. We defined the integral of a non-negative simple A-measurable

function.

• 9. We didn’t quite show that the definition was independent of our

representation of the simple function.

Last week

• 1. We showed that λ∗ is σ-additive on M.
• 2. Hence λ∗|M is a measure! We denote it simply by λ.
• 3. We started on general measure and (a little) integration theory.
• 4. You proved that all measures have nice properties.
• 5. You looked at the concepts of almost everywhere and complete

measures.

• 6. We defined the notion of an A-measurable function.
• 7. We defined a simple A-measurable function.
• 8. We defined the integral of a non-negative simple A-measurable

function.

• 9. We didn’t quite show that the definition was independent of our

representation of the simple function.

Last week

• 1. We showed that λ∗ is σ-additive on M.
• 2. Hence λ∗|M is a measure! We denote it simply by λ.
• 3. We started on general measure and (a little) integration theory.
• 4. You proved that all measures have nice properties.
• 5. You looked at the concepts of almost everywhere and complete

measures.

• 6. We defined the notion of an A-measurable function.
• 7. We defined a simple A-measurable function.
• 8. We defined the integral of a non-negative simple A-measurable

function.

• 9. We didn’t quite show that the definition was independent of our

representation of the simple function.

Last week

• 1. We showed that λ∗ is σ-additive on M.
• 2. Hence λ∗|M is a measure! We denote it simply by λ.
• 3. We started on general measure and (a little) integration theory.
• 4. You proved that all measures have nice properties.
• 5. You looked at the concepts of almost everywhere and complete

measures.

• 6. We defined the notion of an A-measurable function.
• 7. We defined a simple A-measurable function.
• 8. We defined the integral of a non-negative simple A-measurable

function.

• 9. We didn’t quite show that the definition was independent of our

representation of the simple function.

Last week

• 1. We showed that λ∗ is σ-additive on M.
• 2. Hence λ∗|M is a measure! We denote it simply by λ.
• 3. We started on general measure and (a little) integration theory.
• 4. You proved that all measures have nice properties.
• 5. You looked at the concepts of almost everywhere and complete

measures.

• 6. We defined the notion of an A-measurable function.
• 7. We defined a simple A-measurable function.
• 8. We defined the integral of a non-negative simple A-measurable

function.

• 9. We didn’t quite show that the definition was independent of our

representation of the simple function.

Last week

• 1. We showed that λ∗ is σ-additive on M.
• 2. Hence λ∗|M is a measure! We denote it simply by λ.
• 3. We started on general measure and (a little) integration theory.
• 4. You proved that all measures have nice properties.
• 5. You looked at the concepts of almost everywhere and complete

measures.

• 6. We defined the notion of an A-measurable function.
• 7. We defined a simple A-measurable function.
• 8. We defined the integral of a non-negative simple A-measurable

function.

• 9. We didn’t quite show that the definition was independent of our

representation of the simple function.

Last week

• 1. We showed that λ∗ is σ-additive on M.
• 2. Hence λ∗|M is a measure! We denote it simply by λ.
• 3. We started on general measure and (a little) integration theory.
• 4. You proved that all measures have nice properties.
• 5. You looked at the concepts of almost everywhere and complete

measures.

• 6. We defined the notion of an A-measurable function.
• 7. We defined a simple A-measurable function.
• 8. We defined the integral of a non-negative simple A-measurable

function.

• 9. We didn’t quite show that the definition was independent of our

representation of the simple function.

Last week

• 1. We showed that λ∗ is σ-additive on M.
• 2. Hence λ∗|M is a measure! We denote it simply by λ.
• 3. We started on general measure and (a little) integration theory.
• 4. You proved that all measures have nice properties.
• 5. You looked at the concepts of almost everywhere and complete

measures.

• 6. We defined the notion of an A-measurable function.
• 7. We defined a simple A-measurable function.
• 8. We defined the integral of a non-negative simple A-measurable

function.

• 9. We didn’t quite show that the definition was independent of our

representation of the simple function.

Today

• 1. We will show that every non-negative measurable function is an

increasing limit of simple functions.

• 2. We will define the integral of a non-negative measurable function.
• 3. We will look at properties of the integral.
• 4. Notably the Monotone Convergence Theorem.

Today

• 1. We will show that every non-negative measurable function is an

increasing limit of simple functions.

• 2. We will define the integral of a non-negative measurable function.
• 3. We will look at properties of the integral.
• 4. Notably the Monotone Convergence Theorem.

Today

• 1. We will show that every non-negative measurable function is an

increasing limit of simple functions.

• 2. We will define the integral of a non-negative measurable function.
• 3. We will look at properties of the integral.
• 4. Notably the Monotone Convergence Theorem.

Today

• 1. We will show that every non-negative measurable function is an

increasing limit of simple functions.

• 2. We will define the integral of a non-negative measurable function.
• 3. We will look at properties of the integral.
• 4. Notably the Monotone Convergence Theorem.