Convergence in Measure

Let ${\displaystyle (X,{\mathcal {M}},\mu )}$ denote a measure space and let ${\displaystyle f_{n},f:X\to \mathbb {R} }$ for ${\displaystyle n\in \mathbb {N} }$. The sequence ${\displaystyle \{f_{n}\}_{n\in \mathbb {N} }}$ converges to ${\displaystyle f}$ in measure if ${\displaystyle \lim _{n\to \infty }\mu \left(\{x\in X:|f_{n}(x)-f(x)|\geq \epsilon \}\right)=0}$ for any ${\displaystyle \epsilon >0}$. Furthermore, the sequence ${\displaystyle \{f_{n}\}_{n\in \mathbb {N} }}$ is Cauchy in measure if for every ${\displaystyle \epsilon >0,}$ ${\displaystyle \mu (\{x\in X:|f_{n}(x)-f_{m}(x)|\geq \epsilon \})\to 0}$ as ${\displaystyle n,m\to \infty }$ [1]

Properties

• If ${\displaystyle f_{n}\to f}$ in measure and ${\displaystyle g_{n}\to g}$ in measure, then ${\displaystyle f_{n}+g_{n}\to f+g}$ in measure.[2]
• If ${\displaystyle f_{n}\to f}$ in measure and ${\displaystyle g_{n}\to g}$ in measure, then ${\displaystyle f_{n}g_{n}\to fg}$ in measure if this is a finite measure space. [2]

Relation to other types of Convergence

• If ${\displaystyle f_{n}\to f}$ in ${\displaystyle L^{1}(\mu )}$ then ${\displaystyle f_{n}\to f}$ in measure [1]
• If ${\displaystyle f_{n}\to f}$ in measure, then there exists a subsequence ${\displaystyle \{f_{n_{k}}\}_{k\in \mathbb {N} }}$ such that ${\displaystyle f_{n_{k}}\to f}$ almost everywhere.[1]
• If ${\displaystyle \mu (X)<\infty }$ and ${\displaystyle f_{n},f}$ measurable s.t. ${\displaystyle f_{n}\to f}$ almost everywhere Then ${\displaystyle f_{n}\to f}$ in measure.[3]

References

1. Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §2.4
2. Craig, Katy. MATH 201A HW 8. UC Santa Barbara, Fall 2020.
3. Craig, Katy. MATH 201A Lecture 18. UC Santa Barbara, Fall 2020.