Arithmetic of of integrable functions

[steven187]

hello all

i have been trying to prove a property of integrable functions, i had a go at it don't know if it is correct, but I am wondering if there could possibly be a shorter simpler way of proving it alright here we go

$$\int_{a}^{b} f(x)+g(x) dx= \int_{a}^{b} f(x)dx +\int_{a}^{b} g(x) dx$$

My proof

for any partition P of [a,b]

$$U(f+g,P)=\sum_{i=1}^{n}M_i(f+g,P)(x_i-x_{i-1})$$

$$\le \sum_{i=1}^{n}M_i(f,P)(x_i-x_{i-1}) +\sum_{i=1}^{n}M_i(f,P)(x_i-x_{i-1})$$

$$= U(f,P)+U(f,g)$$ similarly

$$L(f+g,P)\ge L(f,P)+L(f,g)$$

there is also partitions $$P_{1}$$ & $$P_{2}$$ of [a,b] such that

$$U(f,P_{1}) <\int_{a}^{b}f+\frac{\epsilon}{2}$$

$$U(g,P_{2}) <\int_{a}^{b}g+\frac{\epsilon}{2}$$

we let $$P=P_{1}UP_{2}$$

$$\int_{a}^{b^U}(f+g)\le U(f+g,P)\le U(f,P)+U(g,P) \le U(f,P_{1})+U(g,P_{2})$$

$$< \int_{a}^{b}f+\frac{\epsilon}{2}+\int_{a}^{b}g+\frac{\epsilon}{2}$$

$$= \int_{a}^{b}f+\int_{a}^{b}g+\epsilon$$
similarly
$$\int_{a_{L}}^{b}(f+g) > \int_{a}^{b}f+\int_{a}^{b}g -\epsilon$$
since all functions are riemann integrable then
$$\int_{a}^{b^U}(f+g) =\int_{a_{L}}^{b}(f+g)= \int_{a}^{b}(f+g)$$
and so it follows that
$$\mid\int_{a}^{b}(f+g)-\int_{a}^{b}f-\int_{a}^{b}g\mid<\epsilon$$ $$\forall\epsilon>0$$

 

[matt grime]

most iof thsat seems unnecessary

if U(f,P)+U(g),P) +> U(f+g,P) => L(f+g,P) => L(f,P)+L(g,P) then the result follows since f and g are integrable

 

[steven187]

hello there

well i have looked at your approach but i don't understand i would appreciate it if you can elaborate,

thank you

 

[steven187]

hello all

also is there any special way of doing such proofs with riemann integrability or is there something I would need to know that is in common with most proofs involving the riemann integral please help its just I am finding it difficult to approach such proofs

thank you

 

[matt grime]

the limits you want to find anre sandwihced between two convergent "sequences". f and g are integrable so the upper and lower sums exist and converge; apply the sandwich principle.

 

[steven187]

hello there

hmm..well I don't get what you mean by limits and convergent sequences, I am getting really confused in how to relates those terms to proving this problem, I want to work on another approach of doing this problem but I don't know where else to start, also is there any special way of doing such proofs with riemann integrability or is there something I would need to know that is in common with most proofs involving the riemann integral please help its just I am finding it difficult to approach such proofs, please help

thank you

 

[matt grime]

there are exactly tow =thnigs to know

1. the definition

2 the criterion of integrability

i have broken my right hand and find typing a pain at the mo'. hopefuly someone else can explain., but you do undrestand that the integral exists if the lower and upper sums converge as the parto=itions longst subpatrition tends to zero and that it suffices to prove this only for a sequence og partitions indexed by N. didn't we prove that for yuou already?

 

[fourier jr]

start wity matt's line. then write down riemann's criterion for integrability, using epsilon/2 for each of f, g. conclude that f + g is riemann integrable by riemann's criterion. then use matt's line & properties of partitions to find out what f + g is.

 

[Hurkyl]

(Assuming the errors in your original post are typos)

You've proven that for any partition P:

U(f + g, P) <= U(f, P) + U(g, P)
and
L(f, P) + L(g, P) <= L(f + g, P)

right?

What do you know about, say, L(f, P) and U(f, P) as P &rarr; &infin;?

 

[steven187]

hello all

You've proven that for any partition P:

U(f + g, P) <= U(f, P) + U(g, P)
and
L(f, P) + L(g, P) <= L(f + g, P)

right?

yep that's what i have proved

What do you know about, say, L(f, P) and U(f, P) as P → ∞?

Well Matt I hope the hand gets better. now from my understanding if the lower and upper sums converge that is as the number of partitions approaches infinity and the subintervals get smaller and smaller, and so if the upper and lower sums converge to the same value then it is riemann integrable.

the only information i can gather is what i have written originally in my proof above and
U(f,P)-L(f,P) <e/2
U(g,P)-L(g,P) <e/2
U(f+g,P)-L(f+g,P) <= U(f,P)-L(f,P)+U(g,P)-L(g,P)<e/2+e/2=e
now from here i don't know what else to do except to go along the same path as the original way i proved it,

 

[Hurkyl]

One thing you can do is take the limit of the inequalities

U(f + g, P) <= U(f, P) + U(g, P)
L(f, P) + L(g, P) <= L(f + g, P)

as |P| → ∞

 

[steven187]

One thing you can do is take the limit of the inequalities

U(f + g, P) <= U(f, P) + U(g, P)
L(f, P) + L(g, P) <= L(f + g, P)

as |P| → ∞

dont you mean |P| → 0
isnt the maximum difference between any two consecutive points of the partition is called the norm or mesh of the partition and is denoted by |P|

if i follow that then

$$\int_{a}^{b}f+\int_{a}^{b}g\le\int_{a_{L}}^{b}(f+g) = \int_{a}^{b}(f+g) =\int_{a}^{b^U}(f+g) \le\int_{a}^{b}f+\int_{a}^{b}g$$

would this be correct, is there anything else that i need to add?

 

[steven187]

hello all

by the way, would i be correct to say that there are only 3 ways to prove one expression is equal to the other and they would be
sandwich theorem a<=b<=a hence a=b
LHS=RHS
|a-b|<e for all e>0 then hence a=b
would there be any other ways? and would anybody know of any links that would explain any other forms of proving inequality?

thanxs