Measure Theory-Lebesgue Measurable

[WannaBe22]

Homework Statement

Let $$A \subseteq R$$ be a Lebesgue-Measurable set. Prove that if the Lebesgue measure of A is less than infinity , then the function $$f(x) = \lambda(A \cap (-\infty,x))$$ is continous.

Homework Equations

The Attempt at a Solution

I'm really confused about the definition of $$\lambda (A)$$ where $$\lambda$$ is the Lebesgue-measure...I've tried taking an $$\epsilon >0$$ and choosing some $$\delta >0$$ for which if $$|x-x_0 | < \delta$$ then $$|f(x)-f(x_0)| <\epsilon$$ but I don't think this is the point...

I'll be delighted to get some guidance

Thanks !

 

[ystael]

Intuitively, $$f(x)$$ is the measure of the portion of $$A$$ "left of $$x$$". So if $$x < x'$$, can you interpret $$f(x') - f(x)$$ in terms of measures?

 

[WannaBe22]

Intuitively, $$f(x')- f(x)$$ is measure of the portion of $$A$$ between $$x$$ and $$x'$$ ... Intuitively , this whole thing seems quite trivial...But when I try to get to the formal aspect of the soloution (as seen in "The attempt at a solution" part) , everything messes out... How can I make the intuition more formal ?
I really hope you'll be able to help me

Thanks !

 

[ystael]

Here are two hints:

1. Given that intuitive description of $$f(x') - f(x)$$, try to come up with an upper bound for $$f(x') - f(x)$$ in terms of $$x' - x$$. This is what you need to prove continuity. (What property of $$A$$ would give the largest possible value for $$f(x') - f(x)$$?)

2. The fact that $$\lambda(A) < \infty$$ is irrelevant to the continuity of $$f$$; you need it merely to define $$f$$.

 

[micromass]

What is f(x)-f(x₀)?

Hint: you must use that if $$A\subseteq B$$ and if those two sets have finite measure, then $$\lambda(B\setminus A)=\lambda(B)-\lambda(A)$$.

 

[WannaBe22]

Thanks a lot! your guidance was very helpful!