Consider the measure space(X,,),(X). Suppose that

f,fn,g,gn:XR,nN

are measurable, i.e, aR:f1({a}),whereisaalgebra.

Prove that a)({xXf(x)>n})0,n. b) ~~ if fnfandgng, then fngnfg.

c) Give an example where(X)=, thenfngn does not converge in measure tofg on X.

REFS (Please refer to the following) Definitions (see Pg 5 for measure) https://www.bauer.uh.edu/rsusmel/phd/sR-0.pdf

Definition of Measure A measure is a nonnegative and \sigma-additive function on a semiring.

Sigma Algebra https://mathworld.wolfram.com/Sigma-Algebra.html

Measurable Functions https://people.math.gatech.edu/~heil/6337/spring11/section3.1.pdf https://sites.ualberta.ca/~rjia/Math417/Notes/chap5.pdf

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Let (X, E, u) be a measure space and functions f, fn : X - R be E-measurable. The function sequence (fn)nen is said to converge to the function f in measure u on X if VE >0: p( x EX|If,(x) - S(2)|2 8}) - 0, n-+ 00. Notation: fn- H f