Lebesgue Measure

Lebesgue integration sucks and you should not ever do it.

>>8267449

Need them for PDEs.

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>>8267442
Anon the best thing you can do is going though several measure and analysis books until you have all figured it out.

>>8267442
you give us an example

>>8267449
Lebesgue integration is beautiful you pleb

>>8267486

I get the idea of finding the length of a given interval, then multiplying 'd' lengths to get the d-dimensional Lebesgue measure for simple sets.. Am I right so far ?

What if there are holes in the interval ? Say (1,4) \ {1.5,2.5,3}. Would the 1D Lebesgue measure be 3 ? Or for example, the set of all x belonging to R such that the 1-norm of x < 3.

How do you calculate the outer/inner measure ?

>>8267442
>calculating Lebesgue measures of simple sets.
>Numerical examples will be appreciated

The Lebesgue measure of [0,1] is 1.

>>8267470
>Need them for PDEs.
>What if there are holes in the interval ? Say (1,4) \ {1.5,2.5,3}.

Go open a book slob you're embarrassing yourself.

>>8267519
Any finite set has measure 0. So where [math]U = (1,4) \setminus \{1.5,2.5,3\}[/math], [math]U \cup \{1.5,2.5,3\} = (1, 4)[/math], so [math]\mu(U) + \mu(\{1.5,2.5,3\}) = \mu((1, 4)) = 3[/math] so [math]\mu(U) = 3 - 0 = 3[/math].

>Or for example, the set of all x belonging to R such that the 1-norm of x < 3.

Well, that's just [math]\{x : |x| < 3\}[/math]. Write it in interval notation and it should be easy.

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>>8267561

Thank you !

>>8267561

Clear explanation. Thanks !