Integration Notation Clarification

Question

I am asked to prove Lebesgue integral's translation invariance property: $$ \int_{R}f(x-h)=\int_Rf(x) \tag{*} $$

1. Is this the same as: $$ \int_{-\infty}^{+\infty}f(x-h) = \int_{-\infty}^{+\infty}f(x)? $$

2. If yes, can I prove this by proving that this is true for any interval in $R$ ?

3. Also, what does it mean if there is no indication of the limits of the integrals in (*)? Does it mean that the equality must be proved for any interval, not just the entire R?

Answer 1

Writing $\displaystyle \int f$ in some contexts means the set over which one integrates is understood from the context, and often it means that it is the whole real line $\mathbb R$. I see this in writing on statistics when one is integrating a product of some function with probability density function to get the expected value of some random variable.

Note that you can write $\mathbb R$ rather than $R$ and that is somewhat conventional.

Also not that I didn't write any $\text{“}dx\text{''}$ in the integral above, but if I'd written $f(x)$ and would have put $dx$ there, indicating which variable is bound. (But binding the variable is far from the whole story of the meaning of that notation.)

Answer 2

1. Yes because $R$ is a poor man's $\Bbb R$.

2. No, it fails on more general intervals. For example, take $f(x)=x,\,R=[0,\,1],\,h=1$ so $\int_Rf(x)dx=\frac12,\,\int_Rf(x-h)dx=-\frac12$.

3. You'll have to deduce from context whether an all-space definite integral (e.g. over $\Bbb R^n$ for an $n$-dimensional $x$) or indefinite integral is intended.

Answer 3

Look at the change of variable : $x'=x-h$ : the domain of integration is untouched, it is still $\mathbb{R}$ and $dx'=dx$, so both integrals have same value.