# Equivalence of integrals

Let $x_1, \ldots, x_n$ be vectors in the normed space $(X, \|\cdot\|)$. Let $\mu$ be the Lebesgue measure on the cube $[-1,1]^n$. Denote vectors in $[-1,1]^n$ by $y=(y_1, \ldots, y_n)$.

Are the following integrals equivalent: $$\int_{[-1,1]^n}\left\|\sum_{i=1}^nx_iy_i\right\|d\mu(y) \quad \mbox{and} \quad \sum_{y\in {\{-1,1\}^n}}\left\|\sum_{i=1}^nx_iy_i\right\|$$

Under 'equivalent' I mean that each integral bounded above by a constant multiple of the other.

• You can get appropriate sizes for the norm bars if you put \left and \right in front of them. – joriki Jul 28 '12 at 18:08
• But I have measure $\mu$ on $[-1, 1]^n$. In other words, can I go from the integration on the boundary of the cube (or even sections of the cube) to the integration on the vertices? – Ali Ual Jul 28 '12 at 18:45
• The second integral should probably be replaced by a sum over the $2^n$ elements of $\{-1,1\}^n$. – Did Jul 28 '12 at 19:03
• Equivalent in the sense of equal or of each bounded above by a constant multiple of the other? – Did Jul 28 '12 at 19:31