Let (X,S, µ) be a measure space such that µ(X) < o. We define d(f, g) = S du. \f-g]| 1+\f-g| Prove that d is a metric on the space of measurable functions, except for the fact that d(f,g) = 0 implies that f = g a.e., not necessarily everywhere. Prove that %3D fn → f in measure as n → ∞ if and only if d(fn, f)→ 0 as n → ∞.
Let (X,S, µ) be a measure space such that µ(X) < o. We define d(f, g) = S du. \f-g]| 1+\f-g| Prove that d is a metric on the space of measurable functions, except for the fact that d(f,g) = 0 implies that f = g a.e., not necessarily everywhere. Prove that %3D fn → f in measure as n → ∞ if and only if d(fn, f)→ 0 as n → ∞.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![|f-8]|
Let (X,S, u) be a measure space such that u(X) < o. We define d(f,8)= SH dµ.
1+|f-g|'
Prove that d is a metric on the space of measurable functions, except for the fact
= g a.e., not necessarily everywhere. Prove that
that d(f,g) = 0 implies that f
fn → f in measure as n → o if and only if d(fn, f) → 0 as n → o.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd677bd59-d83b-4750-814a-bb19e2ec6fd6%2F04a8815e-ad2b-4a06-9c22-87c7a18bc9a8%2Frunidho_processed.png&w=3840&q=75)
Transcribed Image Text:|f-8]|
Let (X,S, u) be a measure space such that u(X) < o. We define d(f,8)= SH dµ.
1+|f-g|'
Prove that d is a metric on the space of measurable functions, except for the fact
= g a.e., not necessarily everywhere. Prove that
that d(f,g) = 0 implies that f
fn → f in measure as n → o if and only if d(fn, f) → 0 as n → o.
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