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Tarea 16, Análisis real

Due November 27

Problem 1

If $latex f_n$ is a sequence of measurable functions, then $latex \{x:\lim f_n \text{ exists} \}$ is a measurable set.

Problem 2

If $latex f:\R\to\R$ is monotone, then is Borel measurable.

Problem 3

If $latex f_n\in L^+$ decreases pointwise to $latex f$, and $latex \int f_1 < \infty$, then $latex \int f = \lim \int f_n$.

Problem 4

Let $latex f_n\in L^1$ such that $latex f_n\rightrightarrows f$.

  1. If $latex \mu(X)<\infty$, then $latex f\in L^1$ and $latex \int f_n\to\int f$.

  2. If $latex \mu(X)=\infty$, then the conclusions of (1.) might fail.

Problem 5

  1. If $latex 1\le p<r\le \infty$, $latex L^p\cap L^r$ is a Banach space with norm $latex ||f||=||f||_p + ||f||_r$.

  2. If $latex 1\le p<q<r\le\infty$, the inclusion map $latex L^p\cap L^r\to L^q$ is continuous.