The answer is **no**, because of fundamental mathematical limitations which originate in set theory regarding [countability][1] (see e.g. [Cantor's theorem][2]) - functions over a given set are more numerous than the set's (power) [cardinality][3]. Neither _Mathematica_ nor any other system can integrate every function in an even much more restricted class; namely, Riemann integrable [functions][4]. All Riemann integrals are equal to Lebesgue integrals if the former are well defined.  The class of functions which could be integrated over a domain in $\mathbb{R}^n$ in _Mathematica_ is of "measure zero" in the class of Lebesgue integrable functions. More precisely, we need [Baire categories][5] to work with general topological concepts of the class of adequate functions.  
When we calculate a definite integral, we are going to think of `NIntegrate` rather than `Integrate`.  

Let's try to integrate a simple Lebesgue integrable function defined in $\mathbb{R}$:

    f[x_] /; x ∈ Rationals && 0 <= x <= 1 := 1
    f[x_] /; ! (x ∈ Rationals) && 0 <= x <= 1 := 0

    f[Sqrt[3]/2]
    f[1/2]
>     0
    1

but neither `Integrate` nor `NIntegrate` can calculate adequate integrals:

    Integrate[ f[x], {x, 0, 1}]
    NIntegrate[ f[x], {x, 0, 1}]

although we know it should be `0`.
Having said that, we can always **supplement** built-in **integration rules** with user-defined ones (see e.g. [Why aren't these additions of integrals and summations equal?](http://mathematica.stackexchange.com/questions/8353/why-arent-these-additions-of-integrals-and-summations-equal)) to **expand a class** of **symbolically** or **numerically** integrable functions - for this purpose _Mathematica_ is most likely the best tool.  
While we could always remedy various problems algorithmically, we shouldn't expect that it could be done in full generality (e.g. because of the finite number of states of computers); otherwise, we should supplement the system's built-in integration rules with **infinitely many** user-defined rules to be able to integrate **every** Lebesgue integrable function.  
Next editions of _Mathematica_ may involve more powerful symbolic capabilities for measure theory and Lebesgue integration problems, but we should realize that there will always be some limitations of the algorithmic approach to integration in the realm of integrable functions. 

**Edit**

The above considerations concern the problem of integration of a possibly wide class of functions. However if one restricts to integration of bi-variate Gaussian density functions there is no need for distinction between Riemann and Lebesgue integrals. Since one needs fast numerical results I'd recommend taking a closer look at the **NIntegrate Integration Strategies** tutorial, especially at [Crude Monte Carlo and Quasi Monte Carlo Strategies](http://reference.wolfram.com/mathematica/tutorial/NIntegrateIntegrationStrategies.html#20795380) and [Global Adaptive Monte Carlo and Quasi Monte Carlo Strategies](http://reference.wolfram.com/mathematica/tutorial/NIntegrateIntegrationStrategies.html#65285686) sections. 

Let's define e.g.

    f1[x_, y_] := PDF[ BinormalDistribution[{1, 3/2}, {1/2, 3/5}, 1/3], {x, y}]
    f2[x_, y_] := PDF[ BinormalDistribution[{4/3, 7/3}, {1, 2/3}, 2/5], {x, y}]

and choose e.g. `τ = 5`.

An especially fast method  would be e.g. `Method -> "AdaptiveMonteCarlo"` with `Boole[ f1[x, y] > 5 f2[x, y]]` - appropriate region selector. Instead of `Boole` we could use `HeavisideTheta`, e.g. `HeavisideTheta[ f1[x, y] - 5 f2[x, y]]` but in this case it appears to be fairly slower (see e.g. [this](http://mathematica.stackexchange.com/questions/39161/how-to-plot-and-find-the-volume-of-a-solid/39164#39164) for the case when it is much faster). Working with `"AdaptiveMonteCarlo"`, one should remember that the method provides a rather rough estimation of the result:

    NIntegrate[ f1[x, y] Boole[ f1[x, y] > 5 f2[x, y]], {x, -∞, ∞}, {y, -∞, ∞}, 
                Method -> "AdaptiveMonteCarlo"]
>     0.370381

A slower but considerably more stable method would be `Method -> "AdaptiveQuasiMonteCarlo"`.

  [1]: http://en.wikipedia.org/wiki/Countable_set
  [2]: http://en.wikipedia.org/wiki/Cantor%27s_theorem
  [3]: http://en.wikipedia.org/wiki/Cardinality
  [4]: http://en.wikipedia.org/wiki/Riemann_integral
  [5]: http://en.wikipedia.org/wiki/Baire_category_theorem