A sequence $f_k:\Omega\rightarrow \mathbb R$ such that $\int f_k=0 \quad \forall k\in \mathbb N $ and $\lim\limits_{k\to\infty} f_k \equiv1$.

Question

Find a sequence of Lebesgue Integrable functions $f_k:\Omega\rightarrow \mathbb R$ such that
$\int f_k=0 \quad \forall k\in \mathbb N $ and $\lim\limits_{k\to\infty} f_k \equiv1$.

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My guess is $$ f_k(x) = sign(x-k) $$ Then integral is $0$ by symmetry and $f_k$ converges to 1. Is it right?
Can you give more examples?

Answer 1

If the functions are supposed to be Lebesgue integrable over $\mathbb R$, then your example, whose absolute value is $1$ everywhere, is not Lebesgue integrable.

An example that illustrates the principle would be $$ f_n(x) = \begin{cases} 1 & x \in [-n, n] \\ -2n & x \in [-(n+1),-n) \\ 0 & \text{else}. \end{cases} $$

Answer 2

If $\Omega=[0,1]$, then you can set $$ f_k(x) = \begin{cases} 1 & x \in [\frac{1}{k}, 1] \\ 1-k & x \in (0,\frac{1}{k}) \\ 1 & x = 0 \end{cases} $$