Which one of the following is true
Let, $f_n(x)$ , for $n\geq 1$, be a sequence of continuous non-negative functions on $[0,1]$ such that $\lim_{n \to \infty} \int_0^1 f_n(x)\,dx = 0$. Which of the following is correct :
A. $f_n \rightarrow 0$ uniformly on $[0,1]$.
B. $f_n$ may not converge uniformly but converges to 0 pointwise.
C. $f_n$ will converge pointwise and the limit may be non-zero.
D. $f_n$ is not guaranteed to have a pointwise limit.
Which one is true?? Please EXPLAIN.
$\endgroup$ 32 Answers
$\begingroup$Take the following example:
$$ f_n(x)=\left\{ \begin{array}{cll} x^n & \text{if} & \text{$n$ odd}, \\ (1-x)^n & \text{if} & \text{$n$ even}. \end{array} \right. $$
Then $f_n(x)\ge 0$, for all $x\in[0,1]$ and $n\in\mathbb N$, and $\int_0^1 f_n(x)\,dx=1/n\to 0$. $\{f_n(x)\}$ converges pointwise to zero, for all $x\in(0,1)$, but it does not converge for $x=0,1$.
Clearly, the answer in D.
$\endgroup$ $\begingroup$$D$ is true. If we didn't need continuous, the standard counter example is the "typewriter function". Let
$f_1 = 1,\\ f_2= \chi_{[0,\frac{1}{2}]},\\ f_3 = \chi_{[\frac{1}{2},1]},\\ f_4 = \chi_{[0,\frac{1}{4}]} \\f_5 =\chi_{[\frac{1}{4},\frac{1}{2}]}\\ \cdots$
and so on. There is no pointwise limit, since given any $x$, then $f(x)$ will be $0$ and $1$ infinitely many times. However, the limit of integrals is $0$, since the integral in this case is just the size of the interval of the characteristic function, which looks like $2^{-n}$
To make this argument work for the continuous function, just consider triangles on these intervals, say the triangle with height $1$ and base $2^{-n}$ going from left to right as the $f_i$ do.
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