Last week 1. We proved the Monotone Convergence Theorem 2. We saw - - PowerPoint PPT Presentation

Last week

• 1. We proved the Monotone Convergence Theorem
• 2. We saw applications of the MCT.
• 3. We proved Fatou’s Lemma.
• 4. We discussed integration of real valued and complex valued functions
• 5. If f : Ω → R is A-measurable then we say f is integrable if
• Ω

|f | dµ < ∞.

• 6. If f is integrable then we define
• Ω

f dµ =

• Ω

f + dµ +

• Ω

f − dµ, where f + and f − denotes the positive and negative parts of f , respectively.

Last week

• 1. We proved the Monotone Convergence Theorem
• 2. We saw applications of the MCT.
• 3. We proved Fatou’s Lemma.
• 4. We discussed integration of real valued and complex valued functions
• 5. If f : Ω → R is A-measurable then we say f is integrable if
• Ω

|f | dµ < ∞.

• 6. If f is integrable then we define
• Ω

f dµ =

• Ω

f + dµ +

• Ω

f − dµ, where f + and f − denotes the positive and negative parts of f , respectively.

Last week

• 1. We proved the Monotone Convergence Theorem
• 2. We saw applications of the MCT.
• 3. We proved Fatou’s Lemma.
• 4. We discussed integration of real valued and complex valued functions
• 5. If f : Ω → R is A-measurable then we say f is integrable if
• Ω

|f | dµ < ∞.

• 6. If f is integrable then we define
• Ω

f dµ =

• Ω

f + dµ +

• Ω

f − dµ, where f + and f − denotes the positive and negative parts of f , respectively.

Last week

• 1. We proved the Monotone Convergence Theorem
• 2. We saw applications of the MCT.
• 3. We proved Fatou’s Lemma.
• 4. We discussed integration of real valued and complex valued functions
• 5. If f : Ω → R is A-measurable then we say f is integrable if
• Ω

|f | dµ < ∞.

• 6. If f is integrable then we define
• Ω

f dµ =

• Ω

f + dµ +

• Ω

f − dµ, where f + and f − denotes the positive and negative parts of f , respectively.

Last week

• 1. We proved the Monotone Convergence Theorem
• 2. We saw applications of the MCT.
• 3. We proved Fatou’s Lemma.
• 4. We discussed integration of real valued and complex valued functions
• 5. If f : Ω → R is A-measurable then we say f is integrable if
• Ω

|f | dµ < ∞.

• 6. If f is integrable then we define
• Ω

f dµ =

• Ω

f + dµ +

• Ω

f − dµ, where f + and f − denotes the positive and negative parts of f , respectively.

Last week

• 1. We proved the Monotone Convergence Theorem
• 2. We saw applications of the MCT.
• 3. We proved Fatou’s Lemma.
• 4. We discussed integration of real valued and complex valued functions
• 5. If f : Ω → R is A-measurable then we say f is integrable if
• Ω

|f | dµ < ∞.

• 6. If f is integrable then we define
• Ω

f dµ =

• Ω

f + dµ +

• Ω

f − dµ, where f + and f − denotes the positive and negative parts of f , respectively.

Last week

• 1. We proved the Monotone Convergence Theorem
• 2. We saw applications of the MCT.
• 3. We proved Fatou’s Lemma.
• 4. We discussed integration of real valued and complex valued functions
• 5. If f : Ω → R is A-measurable then we say f is integrable if
• Ω

|f | dµ < ∞.

• 6. If f is integrable then we define
• Ω

f dµ =

• Ω

f + dµ +

• Ω

f − dµ, where f + and f − denotes the positive and negative parts of f , respectively.

Today

• 1. We will prove the Dominated Convergence Theorem.
• 2. We will prove that if f : [a, b] → C is continuous, then the Riemann

and Lebesgue integrals of f agree, in symbols: b

a

f (x) dx =

• [a,b]

f dµ.

• 3. We will look at some computational examples.

Today

• 1. We will prove the Dominated Convergence Theorem.
• 2. We will prove that if f : [a, b] → C is continuous, then the Riemann

and Lebesgue integrals of f agree, in symbols: b

a

f (x) dx =

• [a,b]

f dµ.

• 3. We will look at some computational examples.

Today

• 1. We will prove the Dominated Convergence Theorem.
• 2. We will prove that if f : [a, b] → C is continuous, then the Riemann

and Lebesgue integrals of f agree, in symbols: b

a

f (x) dx =

• [a,b]

f dµ.

• 3. We will look at some computational examples.