From the given theorem I need the other part of the proof of (c) since the first one is solved. I also need proof of (d). Please, these two tests in the smallest detail, that is, where each step comes from. (DO NOT SHOW THE OTHER ITEMS, ONLY THE PART I REQUESTED OF (C) AND THE (D) COMPLETE.)
From the given theorem I need the other part of the proof of (c) since the first one is solved. I also need proof of (d). Please, these two tests in the smallest detail, that is, where each step comes from. (DO NOT SHOW THE OTHER ITEMS, ONLY THE PART I REQUESTED OF (C) AND THE (D) COMPLETE.)
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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Question
From the given theorem I need the other part of the proof of (c) since the first one is solved. I also need proof of (d). Please, these two tests in the smallest detail, that is, where each step comes from. (DO NOT SHOW THE OTHER ITEMS, ONLY THE PART I REQUESTED OF (C) AND THE (D) COMPLETE.)
![(c)
Proof
Let's consider
2: RXIR → R
(s, t): —> $(s, t): = Sit
f is a continues function.
1 & (s, t)- & (So, to) | = |S+E - (So+to)| = |CS-Sso) + (t-to) |
≤| S-Sol+It-tol
= 11 ( 5, t) - ( So, tollly
1
Let Fix and 9: X-1 & measurable.
For part (b): Re (F): X-1R, Im (F); x->IR; Re(g): X-)|R
and Im (g): X-IR are functions measurable.
I need the proof that fag: x→ ¢ is measurable.
Consider.
F-g = [Re(F) Relg) - Im (f) I m (g)] + i [Re(f) I'm (g) + Im (f) Relig)]
CS Escaneado con CamScanner
Suggestion
-
Show that h: 12 → 122
xh(x) = -x
IS
a measurable function](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff1d184f3-5af6-4008-97aa-5cc87460e1c3%2F44bd8f05-0011-4645-8422-c76f9749a849%2Fhkuhso_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(c)
Proof
Let's consider
2: RXIR → R
(s, t): —> $(s, t): = Sit
f is a continues function.
1 & (s, t)- & (So, to) | = |S+E - (So+to)| = |CS-Sso) + (t-to) |
≤| S-Sol+It-tol
= 11 ( 5, t) - ( So, tollly
1
Let Fix and 9: X-1 & measurable.
For part (b): Re (F): X-1R, Im (F); x->IR; Re(g): X-)|R
and Im (g): X-IR are functions measurable.
I need the proof that fag: x→ ¢ is measurable.
Consider.
F-g = [Re(F) Relg) - Im (f) I m (g)] + i [Re(f) I'm (g) + Im (f) Relig)]
CS Escaneado con CamScanner
Suggestion
-
Show that h: 12 → 122
xh(x) = -x
IS
a measurable function
Transcribed Image Text:Theorem.
Let (X,M) be a measurable space
a) IF U:X-R and V: X → R are measurable functions.
then the complex F:X-¢
IS measurable.
b) IF (:X-> is a measurable function, then the
functions Re(f): X-1R, Im (F): X → R
Y | FI:X-SIR
are measurable.
IF Fizu+iv
X
d. where U: X-IR and V: X-1R then,
Re(f)=U, Im (F) =V and lfl = √√²+v²"
✓
real part of f
↓
↓
modulos of f.
Imaginary part
of f.
CIF fix and g:x- & are measurable functions
then fig: x¢
and fig: x-c
x(F+g) (x) =F(x) +gcx)
are measurable functions.
-) f(x): = u(x) +iv(x)
CS Escaneado con CamScanner
if XeE
x-> (f・g)(x) = f(x)g(x)
d) For ECX, let X₂Cx):= {(1,
Characteristic function of E.
XE X-R15 measurable if and only if, E EM.
Si X&E
be the
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