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Let (X,µ) be a measure space with µ(X) infinite. I'm trying to find an example of a pair (X,µ) and a sequence {f_n} of L^1(X,µ) functions converging uniformly to a function f such that we do not have f_n --> f in L^1(X,µ).
I don't see how you can conclude the statement is false from this.RedX said:If your measure space is Borel then I don't think this statement is true. A sequence of Borel measurable functions that has a limit converges to a Borel function.
morphism said:In fact it is possible to come up with an example satisfying the OP using X=R with Lebesgue measure. Take f_n to be a sequence of rectangles with decreasing height but increasing width, in such a way that f_n -> 0 uniformly but not in L^1. You can even rig it so that f_n doesn't converge in any L^p (1 <= p < inf).
RedX said:Doesn't L^1 mean the space of integrable functions? So why isn't f=0 in this space?
kilimanjaro said:Yes, f(x) = 0 is in L^1. The issue is whether f_n converges to f in the L^1 norm.
L^1 convergence is a type of convergence in mathematical analysis that refers to the convergence of a sequence of functions in the L^1 norm. This norm is defined as the integral of the absolute value of a function over a given interval. In simpler terms, L^1 convergence means that the integral of the difference between a sequence of functions and a limit function approaches zero.
Uniform convergence is a type of convergence in mathematical analysis where a sequence of functions converges to a limit function at a uniform rate. This means that for any given value of x, the difference between the function and the limit function approaches zero as the sequence approaches infinity. In other words, the convergence is independent of the value of x.
L^1 convergence and uniform convergence are both types of convergence that can occur in a sequence of functions. However, they are not the same. In fact, uniform convergence implies L^1 convergence, but the converse is not always true. This means that if a sequence of functions has a uniform convergence, it also has an L^1 convergence. However, a sequence of functions may have an L^1 convergence without having a uniform convergence.
Pointwise convergence and uniform convergence are two types of convergence that can occur in a sequence of functions. Pointwise convergence means that for each value of x, the sequence of functions converges to the limit function. On the other hand, uniform convergence means that the convergence occurs at a uniform rate, independent of the value of x. This means that for any given value of x, the difference between the function and the limit function approaches zero at the same rate as the sequence approaches infinity.
L^1 convergence and L^2 convergence are two different types of convergence in mathematical analysis. L^1 convergence refers to the convergence of a sequence of functions in the L^1 norm, which is defined as the integral of the absolute value of a function. On the other hand, L^2 convergence refers to the convergence of a sequence of functions in the L^2 norm, which is defined as the integral of the square of a function. In simpler terms, L^1 convergence measures the total difference between a sequence of functions and a limit function, while L^2 convergence measures the squared difference between them.