Let \(\left\{f_{n}(x)ight\}_{n=1}^{\infty}\) be a sequence of real functions defined by \(f_{n}(x)=(1+\) \(\left.n^{-1}ight) \delta\{x ;(0,1)\}\) for all \(n \in \mathbb{N}\).

a. Prove that

\[\lim _{n ightarrow \infty} f_{n}(x)=\delta\{x ;(0,1)\}\]

for all \(x \in \mathbb{R}\), and hence conclude that

\[\int_{-\infty}^{\infty} \lim _{n ightarrow \infty} f_{n}(x) d x=1\]

b. Compute

\[\lim _{n ightarrow \infty} \int_{-\infty}^{\infty} f_{n}(x) d x\]

Does this match the result you found above?

c. State whether Theorem 1.11 applies to this case, and use it to explain the results you found above.

Transcribed Image Text:

Theorem 1.11 (Lebesgue). Let {fn(x)}1 be a sequence of real functions. pw Suppose that fnf as n for some real valued function f, and that there exists a real function g such that L 9(x)\dx < , and fn(x)g(x) for all x R and n = N. Then L |_ |f(x)\dx < , and lim n Lo fn(x)dx = L lim n fn(x)dx = f(x)dr.