LEBESGUE MEASURE

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Ser Peyep

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Disclaimer: This video discusses about the Lebesgue measure and is divided into five sub parts: 1. Important terms and definition of measurable set (01:00), 2. Properties/Theorems related to Lebesgue measure (03:40), 3. Important of Lebesgue measure (11:00), 4. Articles related to Lebesgue measure (13:40), and 5. Analogy of Lebesgue measure in Real life (16:00). We give emphasis on the definition of theorems but proofs are not included.

While the outer measure has the advantage that it is defined for all sets, it is not countably additive. It becomes countably additive, however, if we suitable reduce the family of sets on which it is defined. Perhaps the best way of doing this is to use the following definition due to Carathéodory:

Definition: Any set of E is Lebesgue measurable if for all A subset of R,
m*(A) = m∗ (A intersection E)+ m∗ (A intersection E-complement). This is called the Carathéodory splitting condition.

We give emphasis to these 5 theorems/properties, and the rest are extensions.

Theorems:

If m*E = 0, then E is measurable.

Proof:
Let A be any set of real numbers. Then, A intersection E ⊆ E ⇒ m∗ (A intersection E) ≤ m∗ (E) and A intersection Ec ⊆ A ⇒ m∗ (A intersection Ec) ≤ m∗ (A).
Therefore, m ∗ ( A intersection E) + m ∗ ( A intersection Ec) ≤ m ∗ (E) + m ∗ (A), = 0 + m ∗ (A). Hence, E is measurable.

If E1 and E2 are measurable, so is E1 ∪ E2.

Proof:
Let A be any set of real numbers and E1, E2 be two measurable sets. Since E2 is measurable, we have m*(A intersection E1c) = m*(A intersection E1c intersection E2c) + m*(A intersection E1c intersection E2).

Now, A intersection (E1 union E2) = [A intersection E1] union [A intersection E2]
= [A intersection E1] union [A intersection E2 intersection E1c]
= m* (A intersection (E1 union E2)) <= m* (A intersection E1) + m* [A intersection E2 intersection E1c]

Let us now consider,

m* (A intersection (E1 union E2)) + m* (A intersection (E1 union E2)c) <= m* (A intersection E1) + m* [A intersection E1c] = m* (A)
Since E1 is also given measurable, hence E1 intersection E2 is also measurable.

The family M of measurable sets is an algebra of sets.
The family M of measurable sets is a σ-algebra.
The interval (a, ∞) is measurable.

Source of Article:

SerPeyep

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