Existence of a certain function

Does there exist a continuous function $f:\mathbb{R}\to(0,\infty)$ that satisfies

$$\int_{\mathbb{R}} f^n d L^1 = 1$$

for every natural $n$? ($L^1$ is Lebesgue measure on $\mathbb{R}$)

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uhmmmm... continues. It's not very continues in 0 and 1, is it? – user19502 Feb 8 '13 at 21:01
Do you have any thoughts/attempts to share? – Jonas Meyer Feb 8 '13 at 21:02
I tried with $e^{-x^2}$, and I can construct a series of functions, $f_n$, that satisfy $\int_{\mathbb{R}} f_n^n dx=1$:$$\int_{-\infty }^{\infty } \left(\sqrt[2n]{\frac{n}{\pi }}e^{-x{}^{\wedge}2}\right)^n \, dx,$$, but I cannot think of one specific function – user19502 Feb 8 '13 at 21:03

No. Suppose such $f$ does exist to reach a contradiction. Then $\sup\limits_{x\in \mathbb R}f(x)=\lim\limits_{n\to\infty}\left(\int_{\mathbb R} f^n dL\right)^{1/n}=1$, so $f(x)\leq 1$ for all $x$. Hence $f-f^2$ is a nonnegative function, and by continuity $f-f^2$ is not always zero. This implies (again using continuity) that $\int_{\mathbb R} f-f^2 dL>0$.
And with a bit more work, you can prove that any measurable function with $\int f^n\ dx = 1$ for all positive integers $n$ is (almost everywhere) the indicator function of a set of measure $1$. – Robert Israel Feb 8 '13 at 21:15