Give an example of a function that satisfies strict inequality in Fatou's lemma and ....

Question

Find a sequence of functions $f_n\ge 0 $ that is continuous on the interval $[0,1]$ satisfies $\int f_n dm \leq 1$ and strict inequality occur in Fatou's lemma.

I could find one that satisfies the strict inequality but discontinuous. example characteristics function $f_n(x)=\large{X}_{[n,n+1]}(x)$. but I need one that is continuous as well and satisfies all the above. any assisstance is appreciated.

Answer 1

You should be able to make $X_[n,n+1]$ continuous by shrinking the interval where your function is exactly one, and connecting the points in a small epsilon neighborhood of $n$ and $n+1$, like so

In this way, the integral is $<1$, but the function becomes continuous.

The "rounding out" that I was referring to was to make the function differentiable, which is not needed in your question. I apologize for misreading, and for my horrible drawing :)

Answer 2

Try the sequence $f_n(x) = n^2x^n(1-x).$