Yes, a subsequence of measurable functions can converge in L1. This means that the sequence of functions converges in the L1-norm, which is a measure of distance between functions in the space of integrable functions.
The L1-norm is a measure of distance between functions in the space of integrable functions. It is defined as the integral of the absolute value of a function over its domain.
No, not all measurable functions are integrable. A function is considered integrable if its L1-norm is finite, meaning that it is possible to calculate the integral of the absolute value of the function over its domain.
Convergence in L1 is a stronger form of convergence than pointwise convergence. Pointwise convergence only requires that a sequence of functions approaches a limit at each point in the domain, while convergence in L1 requires that the L1-norm of the difference between the limit and the functions in the sequence approaches zero.
Yes, there are several applications of convergence in L1 in mathematics and statistics. It is commonly used in the study of Fourier series and in the proof of the Central Limit Theorem. It also has applications in probability theory, where it is used to define the concept of convergence in distribution.